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We consider the Dirichlet-to-Neumann operator N associated with a general elliptic operator A u = - ∑ k , l = 1 d ∂ k (c kl ∂ l u) + ∑ k = 1 d (c k ∂ k u - ∂ k (b k u)) + c 0 u ∈ D ′ (Ω) with possibly complex coefficients. We study three problems: (1) Boundedness on C ν and on L p of the commutator [ N , M g ] , where M g denotes the multiplication operator by a smooth function g. (2) Hölder and L p -bounds for the harmonic lifting associated with A . (3) Poisson bounds for the heat kernel of N . We solve these problems in the case where the coefficients are Hölder continuous and the underlying domain is bounded and of class C 1 + κ for some κ > 0 . For the Poisson bounds we assume in addition that the coefficients are real-valued. We also prove gradient estimates for the heat kernel and the Green function G of the elliptic operator with Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the existence of nontrivial nonnegative solutions for a (p , N) -Laplacian Schrödinger–Kirchhoff problem in ℝ N with singular exponential nonlinearity. The main features of the paper are the (p , N) growth of the elliptic operators, the double lack of compactness, and the fact that the Kirchhoff function is of degenerate type. To establish the existence results, we use the mountain pass theorem, the Ekeland variational principle, the singular Trudinger–Moser inequality, and a completely new Brézis–Lieb-type lemma for singular exponential nonlinearity. [ABSTRACT FROM AUTHOR]

In this paper, we consider an elliptic operator with constant coefficients and we estimate the maximal function of the tangential gradient of the kernel of the double layer potential with respect to its first variable. As a consequence, we deduce the validity of a continuity property of the double layer potential in Hölder spaces on the boundary that extends previous results for the Laplace operator and for the Helmholtz operator. [ABSTRACT FROM AUTHOR]

We propose a third-order numerical integrator based on the Neumann series and the Filon quadrature, designed mainly for highly oscillatory partial differential equations. The method can be applied to equations that exhibit small or moderate oscillations; however, counter-intuitively, large oscillations increase the accuracy of the scheme. With the proposed approach, the convergence order of the method can be easily improved. Error analysis of the method is also performed. We consider linear evolution equations involving first- and second-time derivatives that feature elliptic differential operators, such as the heat equation or the wave equation. Numerical experiments consider the case in which the space dimension is greater than one and confirm the theoretical study. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the existence of nontrivial nonnegative solutions for a (p , N) -Laplacian Schrödinger–Kirchhoff problem in ℝ N with singular exponential nonlinearity. The main features of the paper are the (p , N) growth of the elliptic operators, the double lack of compactness, and the fact that the Kirchhoff function is of degenerate type. To establish the existence results, we use the mountain pass theorem, the Ekeland variational principle, the singular Trudinger–Moser inequality, and a completely new Brézis–Lieb-type lemma for singular exponential nonlinearity. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence, uniqueness and qualitative properties of heteroclinic solution for a class of autonomous quasilinear ordinary differential equations of the Allen–Cahn type given by −ϕ(|u′|)u′′ + V′(u) = 0in ℝ, where V is a double-well potential with minima at t = ±α and ϕ : (0, +∞) → (0, +∞) is a C1 function satisfying some technical assumptions. Our results include the classic case ϕ(t) = tp−2, which is related to the celebrated p-Laplacian operator, presenting the explicit solution in this specific scenario. Moreover, we also study the case ϕ(t) = 1 1+t2, which is directly associated with the prescribed mean curvature operator. [ABSTRACT FROM AUTHOR]

In this article, we present a numerical scheme for solving a time-fractional stochastic Cahn–Hilliard equation driven by an additive fractionally integrated Gaussian noise. The model involves a Caputo fractional derivative in time of order α ∈ (0 , 1) and a fractional time-integral noise of order γ ∈ [ 0 , 1 ]. Our numerical approach combines a piecewise linear finite element method in space with a convolution quadrature in time, designed to handle both time-fractional operators, along with the L 2 -projection for the noise. We conduct a detailed analysis of both spatially semidiscrete and fully discrete schemes, deriving strong convergence rates through energy-based arguments. The solution's temporal Hölder continuity played a key role in the error analysis. Unlike the stochastic Allen–Cahn equation, the inclusion of the unbounded elliptic operator in front of the cubic nonlinearity in our model added complexity and challenges to the error analysis. To address these challenges, we introduce novel techniques and refined error estimates. We conclude with numerical examples that validate our theoretical findings. [ABSTRACT FROM AUTHOR]

Determining the unknown order of the fractional derivative in differential equations simulating various processes is an important task of modern applied mathematics. In the last decade, this problem has been actively studied by specialists. A number of interesting results with a certain applied significance were obtained. This paper provides a short overview of the most interesting works in this direction. Next, we consider the problem of determining the order of the fractional derivative in the subdiffusion equation, provided that the elliptic operator included in this equation has at least one negative eigenvalue. An asymptotic formula is obtained according to which, knowing the solution at least at one point of the domain under consideration, the required order can be calculated. [ABSTRACT FROM AUTHOR]

Let L be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients on R n and X a ball quasi-Banach function space on R n satisfying some mild assumptions. Denote by H X , L (R n) the Hardy space, associated with both L and X, which is defined via the Lusin area function related to the semigroup generated by L. In this article, the authors establish both the maximal function and the Riesz transform characterizations of H X , L (R n) . The results obtained in this article have a wide range of generality and can be applied to the weighted Hardy space, the variable Hardy space, the mixed-norm Hardy space, the Orlicz–Hardy space, the Orlicz-slice Hardy space, and the Morrey–Hardy space, associated with L. In particular, even when L is a second order divergence form elliptic operator, both the maximal function and the Riesz transform characterizations of the mixed-norm Hardy space, the Orlicz-slice Hardy space, and the Morrey–Hardy space, associated with L, obtained in this article, are completely new. [ABSTRACT FROM AUTHOR]

This article establishes the boundary Hölder continuity of stable solutions to semilinear elliptic problems in the optimal range of dimensions n ≤ 9 , for C 1 , 1 domains. We consider equations − L ⁢ u = f ⁢ (u) in a bounded C 1 , 1 domain Ω ⊂ R n , with u = 0 on ∂ Ω , where 퐿 is a linear elliptic operator with variable coefficients and f ∈ C 1 is nonnegative, nondecreasing, and convex. The stability of 푢 amounts to the nonnegativity of the principal eigenvalue of the linearized equation − L − f ′ ⁢ (u) . Our result is new even for the Laplacian, for which [X. Cabré, A. Figalli, X. Ros-Oton and J. Serra, Stable solutions to semilinear elliptic equations are smooth up to dimension 9, Acta Math.224 (2020), 2, 187–252] proved the Hölder continuity in C 3 domains. [ABSTRACT FROM AUTHOR]

Let Ω ⊆ R d be open, A a complex uniformly strictly accretive d × d matrix-valued function on Ω with L ∞ coefficients, b and c two d-dimensional vector-valued functions on Ω with L ∞ coefficients and V a locally integrable nonnegative function on Ω . Consider the operator L A , b , c , V = - div (A ∇ ·) + ∇ , b ¯ - div (c ·) + V with mixed boundary conditions on Ω . We extend the bilinear inequality that Carbonaro and Dragičević proved in [Bilinear embedding for Schrödinger-type operators with complex coefficients. Publ. Mat. (to appear)] in the special cases when b = c = 0 , previously proved in (Calc Var Part Differ Equ 59(3):36, Paper No. 104, 2020) when V = 0 as well. As a consequence, we obtain that the solution to the parabolic problem u ′ (t) + L A , b , c , V u (t) = f (t) , u (0) = 0 , has maximal regularity in L p (Ω) , for all p > 1 such that A satisfies the p-ellipticity condition that Carbonaro and Dragičević introduced in (J Eur Math Soc 22(10):3175–3221, 2020) and b, c, V satisfy another condition that we introduce in this paper. Roughly speaking, V has to be “big” with respect to b and c. We do not impose any conditions on Ω , in particular, we do not assume any regularity of ∂ Ω , nor the existence of a Sobolev embedding. [ABSTRACT FROM AUTHOR]

For a class of hypoelliptic operators with real-analytic coefficients, we provide a criterion ensuring a partial analyticity result. As a consequence, even when the "elliptic" strong unique continuation (i.e. a solution of the homogeneous equation which vanishes of infinite order at a point is zero near such a point) fails, a weaker form of "propagation" of zeroes still holds. [ABSTRACT FROM AUTHOR]

We prove an existence (and regularity) result of weak solutions u ∈ W 0 1 , p (Ω) ∩ W loc 1 , q (Ω) , to a Dirichlet problem for a second order elliptic equation in divergence form, under general and p , q − growth conditions of the differential operator. This is a first attempt to extend to general growth the well known Leray-Lions existence theorem, which holds under the so-called natural growth conditions with q = p. We found a way to treat the general context with explicit dependence on (x , u) , other than on the gradient variable ξ = D u ; these aspects require particular attention due to the p , q -context, with some differences and new difficulties compared to the standard case p = q. [ABSTRACT FROM AUTHOR]

In $ L_2(\mathbb{R}^d) $ L 2 (R d) , we consider an elliptic differential operator $ \mathcal{A}_\varepsilon $ A ϵ , $ \varepsilon \gt 0 $ ϵ > 0 , of the form $ \mathcal {A}_\varepsilon = - \operatorname {div} g(\mathbf {x}/\varepsilon) \nabla + \varepsilon ^{-2} V(\mathbf {x}/\varepsilon) $ A ϵ = − div ⁡ g (x / ϵ) ∇ + ϵ − 2 V (x / ϵ) with periodic coefficients. For hyperbolic equations with the operator $ \mathcal {A}_\varepsilon $ A ϵ , analogs of homogenization problems related to an arbitrary point of the dispersion relation of the operator $ \mathcal {A}_1 $ A 1 are studied (the so-called high-frequency homogenization). For the solutions of the Cauchy problems for these equations with special initial data, approximations in $ L_2(\mathbb {R}^d) $ L 2 (R d) -norm for small ε are obtained. [ABSTRACT FROM AUTHOR]

This paper is devoted to the extension of a quasi-Newton/variable preconditioning (QNVP) method to non-smooth problems, motivated by elasto-plastic models. Two approaches are discussed: the first one is carried out via regularized approximations of the nonsmooth problem, and the second one gives an extension to nonsmooth operators in order to be applied directly. Convergence analysis is presented for both variants. Then these abstract methods are applied to elasto-plasticity where two different variants of QNVP are investigated and combined with the deflated conjugate gradient and aggregation-based algebraic multigrid methods. The convergence results are illustrated on numerical examples in 3D inspired by real-life problems, and they demonstrate that the suggested QNVP methods are competitive with the standard Newton method. Well-documented Matlab codes on elasto-plasticity are used and enriched by the suggested methods. • Extension of the quasi-Newton/variable preconditioning method for non-smooth operators. • Convergence analysis of the method and its application to elasto-plasticity. • Combination of the method with the deflated conjugate gradient and aggregation-based algebraic multigrid methods. • Extension of well-documented and available Matlab codes on elasto-plasticity. • Comparison of the quasi-Newton and Newton methods on 3D numerical examples. [ABSTRACT FROM AUTHOR]

We consider nonconstant periodic constrained minimizers of semilinear elliptic equations for integro-differential operators in ℝ. We prove that, after an appropriate translation, each of them is necessarily an even function which is decreasing in half its period. In particular, it has only two critical points in half its period, the absolute maximum and minimum. If these statements hold for all nonconstant periodic solutions, and not only for constrained minimizers, remains as an open problem. Our results apply to operators with kernels in two different classes: kernels K which are convex and kernels for which K(τ1∕2) is a completely monotonic function of τ. This last new class arose in our previous work on nonlocal Delaunay surfaces in ℝn. Due to their symmetry of revolution, it gave rise to a 1d problem involving an operator with a nonconvex kernel. Our proofs are based on a not so well-known Riesz rearrangement inequality on the circle 1 established in 1976. We also put in evidence a new regularity fact which is a truly nonlocal-semilinear effect and also occurs in the nonperiodic setting. Namely, for nonlinearities in Cβ and when 2 s + β < 1 (2s being the order of the operator), the solution is not always C 2 s + β − ϵ for all ϵ > 0. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Mathematical Physics, Analysis, Geometry (18129471) is the property of B Verkin Institute for Low Temperature Physics & Engineering and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Journal of Mahani Mathematical Research Center is the property of Shahid Bahonar University of Kerman, Department of Pure Mathematics and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We consider a class of nonvariational degenerate elliptic operators of the kind L u = ∑ i , j = 1 m a i j (x) X i X j u where { a i j (x) } i , j = 1 m is a symmetric uniformly positive matrix of bounded measurable functions defined in the whole R n (n > m), possibly discontinuous but satisfying a VMO assumption, and X 1 ,... , X m are real smooth vector fields satisfying Hörmander rank condition in the whole R n and 1-homogeneous w.r.t. a family of nonisotropic dilations. We do not assume that the vector fields are left invariant w.r.t. an underlying Lie group of translations. We prove global W X 2 , p a-priori estimates, for every p ∈ (1 , ∞) , of the kind: ‖ u ‖ W X 2 , p (R n) ≤ c { ‖ L u ‖ L p (R n) + ‖ u ‖ L p (R n) } for every u ∈ W X 2 , p (R n). We also prove higher order estimates and corresponding regularity results: if a i j ∈ W X k , ∞ (R n) , u ∈ W X 2 , p (R n) , L u ∈ W X k , p (R n) , then u ∈ W X k + 2 , p (R n) and ‖ u ‖ W X k + 2 , p (R n) ≤ c { ‖ L u ‖ W X k , p (R n) + ‖ u ‖ L p (R n) }. [ABSTRACT FROM AUTHOR]

In this paper, we consider the principal spectral theory for the nonlocal dispersal eigenvalue problem (1) d ρ p [ ∫ Ω J ρ (x − y) u (y) d y − u (x) ] + κ ρ q [ ∫ Ω G ρ (x − y) u (y) d y − u (x) ] + a (x) u (x) = − λ u (x) , x ∈ Ω ¯ , where Ω ⊂ R N is a bounded smooth domain, J ρ (x) = ρ N J (ρ x) , G ρ (x) = ρ N G (ρ x) , the kernel functions J (x) and G (x) are nonnegative, J (x) is symmetric, the parameter ρ is positive, d , κ are positive, p , q are given constants, and a ∈ C (Ω ¯). We investigate the limiting behavior of the principal spectral point or generalized principal eigenvalue of (1) as ρ → ∞ and ρ → 0. When ρ ≫ 1 , the nonlocal dispersal operator u (⋅) ↦ d ρ p (∫ Ω J ρ (⋅ − y) u (y) d y − u (⋅)) + κ ρ q (∫ Ω G ρ (⋅ − y) u (y) d y − u (⋅)) + a (⋅) u (⋅) with ν 0 ≠ 0 behaves like the elliptic operator u (⋅) ↦ d c 0 ρ p − 2 Δ u − κ ρ q − 1 ν 0 ⋅ ∇ u + a (⋅) u (⋅) on Ω with Dirichlet boundary condition on ∂Ω, where c 0 = 1 2 N ∫ R N J (y) | y | 2 d y , ν 0 = 〈 ∫ R N G (x) x 1 d x , ∫ R N G (x) x 2 d x , ⋯ , ∫ R N G (x) x N d x 〉. The results obtained in the paper are novel when the dispersal kernel function G (x) is asymmetric. Moreover, when G is asymmetric with ν 0 ≠ 0 , and p ≥ 2 , q > p − 1 or p < 2 , q > 1 , it is seen that the limiting behavior of the generalized principal eigenvalue of (1) as ρ → ∞ is strikingly different from the limiting behavior of the principal eigenvalue of the elliptic operator u (⋅) ↦ d c 0 ρ p − 2 Δ u − κ ρ q − 1 ν 0 ⋅ ∇ u + a (⋅) u (⋅) on Ω with Dirichlet boundary condition on ∂Ω. The main results are used to study the asymptotic dynamics of nonlinear nonlocal dispersal problems. [ABSTRACT FROM AUTHOR]

The topological index of elliptic operators on a two-dimensional manifold with cylindrical ends is constructed in terms of periodic cyclic cohomology of the algebra of symbols of these operators. The topological index for operators with shifts is constructed in the same way. [ABSTRACT FROM AUTHOR]

The authors Matsaev and Mogulskii identified a wide class of weak perturbations of a positive compact operator that have no nonzero eigenvalues, i.e., are Volterra operators. By a weak perturbation of a positive operator we mean an operator of the form , where is a compact operator such that is continuously invertible. On the other hand, these weak perturbations have a complete system of root vectors if the self-adjoint operator belongs to a von Neumann–Schatten class. In this paper, we consider compact operators that can be represented as the sum of two compact operators (i.e., is not necessarily a weak perturbation), where is a positive operator. In this paper, we prove theorems on the existence of nonzero eigenvalues for such operators. As is known, Cauchy problems for differential equations are, as a rule, well-posed Volterra problems. However, Hadamard's example shows that the Cauchy problem for the Laplace equation is ill posed. Up to now, not a single Volterra well-posed restriction or extension is known for an elliptic-type equation. Thus, the following question arises: "Does there exist a Volterra well-posed restriction of the maximal operator or a Volterra well-posed extension of the minimal operator generated by elliptic-type equations?" The abstract theorems on the existence of eigenvalues obtained here show that a wide class of well-posed restrictions of the maximal operator and a wide class of well-posed extensions of the minimal operator generated by elliptic-type equations cannot be Volterra operators. Moreover, in the two-dimensional case, it is proved that, for the Laplace operator, there are no well-posed Volterra restrictions and extensions at all. [ABSTRACT FROM AUTHOR]

We prove new multiplicity results for some nonlocal critical growth elliptic problems in bounded domains. More specifically, we show that the problems considered here have arbitrarily many solutions for all sufficiently large values of a certain parameter λ>0$\lambda > 0$. In particular, the number of solutions goes to infinity as λ→∞$\lambda \rightarrow \infty$. We also give an explicit lower bound on λ$\lambda$ in order to have a given number of solutions. This lower bound is in terms of a sequence of eigenvalues of the associated nonlocal elliptic operator. The proofs are based on an abstract critical point theorem. [ABSTRACT FROM AUTHOR]

In this paper, we explore the ball convergence properties of enhanced Phragmén–Lindelöf type methods for solving the Dirichlet problem with an elliptic operator. By placing requirements on the elliptic operator and auxiliary points, we characterize the ball limit of a series of elliptic operators in non‐divergence form. Considering all dimensions, it becomes apparent that dealing solely with measurable and bounded coefficients for this class of operators is insufficient, necessitating additional regularity assumptions on them. We find the radii of convergence balls using recurrence relations, so as to guarantee the convergence of iterative reconstruction techniques, beginning from any point inside the ball centered on an auxiliary point. [ABSTRACT FROM AUTHOR]

We discuss in which cases the Dunkl convolution u∗kv$u*_kv$ of distributions u,v$u,v$, possibly both with non‐compact support, can be defined and study its analytic properties. We prove results on the (singular‐)support of Dunkl convolutions. Based on this, we are able to prove a theorem on elliptic regularity for a certain class of Dunkl operators, called elliptic Dunkl operators. Finally, for the root system An−1$A_{n-1}$ we consider the Riesz distributions (Rα)α∈C$(R_\alpha)_{\alpha \in \mathbb {C}}$ and prove that their Dunkl convolution exists and that Rα∗kRβ=Rα+β$R_\alpha *_kR_\beta = R_{\alpha +\beta }$ holds. [ABSTRACT FROM AUTHOR]

In this work we study the following nonlinear anisotropic elliptic equations (P) f - Σn=1 ∂χi(∣∂χiu|pi(x)-2dxiu) + ∣u∣pm(x)-2u = f (x,u) in Ω Σi=1 ∣∂χi u|pi(x)-2dxi u.νi + e(x)|u|Pm(x)-2u = 0 on ∂Ω. We set up that the problem (P) admits a sequence of weak solutions and multiplicity result under suitable conditions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the first eigenvalue for one‐dimensional elliptic operators with killing. Two‐sided approximation procedures and basic estimates of the first eigenvalue are given in both the half line and the whole line. The proofs are based on the h$h$‐transform, Chen's dual variational formulas, and the split technique. In particular, a few examples are presented to illustrate the power of our results. [ABSTRACT FROM AUTHOR]

In this paper, we study the eta-invariant of elliptic parameter-dependent boundary-value problems and its main properties. Using Melrose's approach, we define the eta-invariant as a regularization of the winding number of the family. In this case, the regularization of the trace requires obtaining the asymptotics of the trace of compositions of invertible parameter-dependent boundaryvalue problems for large values of the parameter. Obtaining the asymptotics uses the apparatus of pseudodifferential boundary-value problems and is based on the reduction of parameter-dependent boundary-value problems to boundary-value problems with no parameter. [ABSTRACT FROM AUTHOR]

We show weighted non-autonomous L q (L p) maximal regularity for families of complex second-order systems in divergence form under a mixed regularity condition in space and time. To be more precise, we let p , q ∈ (1 , ∞) and we consider coefficient functions in C t β + ε with values in C x α + ε subject to the parabolic relation 2 β + α = 1. If p < d α , we can likewise deal with spatial H x α + ε , d α regularity. The starting point for this result is a weak (p , q) -solution theory with uniform constants. Further key ingredients are a commutator argument that allows us to establish higher a priori spatial regularity, operator-valued pseudo differential operators in weighted spaces, and a representation formula due to Acquistapace and Terreni. Furthermore, we show p -bounds for semigroups and square roots generated by complex elliptic systems under a minimal regularity assumption for the coefficients. [ABSTRACT FROM AUTHOR]

Let n ≥ 2 and Ω be a bounded Lipschitz domain of R n . Assume that L R is a second-order divergence form elliptic operator having real-valued, bounded, symmetric, and measurable coefficients on L 2 (Ω) with the Robin boundary condition. In this article, via first obtaining the Hölder estimate of the heat kernels of L R , the authors establish a new atomic characterization of the Hardy space H L R p (Ω) associated with L R . Using this, the authors further show that, for any given p ∈ (n n + δ 0 , 1 ] , H z p (Ω) + L ∞ (Ω) = H L N p (Ω) = H L R p (Ω) ⫋ H L D p (Ω) = H r p (Ω) , where H L D p (Ω) and H L N p (Ω) denote the Hardy spaces on Ω associated with the corresponding elliptic operators respectively having the Dirichlet and the Neumann boundary conditions, H z p (Ω) and H r p (Ω) respectively denote the "supported type" and the "restricted type" Hardy spaces on Ω , and δ 0 ∈ (0 , 1 ] is the critical index depending on the operators L D , L N , and L R . The authors then obtain the boundedness of the Riesz transform ∇ L R - 1 / 2 on the Lebesgue space L p (Ω) when p ∈ (1 , ∞) [if p > 2 , some extra assumptions are needed] and its boundedness from H L R p (Ω) to L p (Ω) when p ∈ (0 , 1 ] or to H r p (Ω) when p ∈ (n n + 1 , 1 ] . As applications, the authors further obtain the global regularity estimates, in L p (Ω) when p ∈ (0 , p 0) and in H r p (Ω) when p ∈ (n n + 1 , 1 ] , for the inhomogeneous Robin problem of L R on Ω , where p 0 ∈ (2 , ∞) is a constant depending only on n, Ω , and the operator L R . The main novelties of these results are that the range (0 , p 0) of p for the global regularity estimates in the scale of L p (Ω) is sharp and that, in some sense, the space X : = H L R 1 (Ω) is also optimal to guarantee both the boundedness of ∇ L R - 1 / 2 from X to L 1 (Ω) or to H r 1 (Ω) and the global regularity estimate ‖ ∇ u ‖ L n n - 1 (Ω ; R n) ≤ C ‖ f ‖ X for inhomogeneous Robin problems with C being a positive constant independent of both u and f. [ABSTRACT FROM AUTHOR]

On a compact Riemannian manifold M with boundary Y, we express the log of the zeta-determinant of the Dirichlet-to-Neumann operator acting on q-forms on Y as the difference of the log of the zeta-determinant of the Laplacian on q-forms on M with the absolute boundary condition and that of the Laplacian with the Dirichlet boundary condition with an additional term which is expressed by curvature tensors. When the dimension of M is 2 and 3, we compute these terms explicitly. We also discuss the value of the zeta function at zero associated to the Dirichlet-to-Neumann operator by using a metric rescaling method. As an application, we recover the result of the conformal invariance obtained in Guillarmou and Guillope (Int Math Res Not IMRN 2007(22):rnm099, 2007) when dim M = 2 . [ABSTRACT FROM AUTHOR]

Stability and optimal convergence analysis of a non-uniform implicit-explicit L1 finite element method (IMEX-L1-FEM) is studied for a class of time-fractional linear partial differential/integro-differential equations with non-self-adjoint elliptic part having (space-time) variable coefficients. The proposed scheme is based on a combination of an IMEX-L1 method on graded mesh in the temporal direction and a finite element method in the spatial direction. With the help of a discrete fractional Grönwall inequality, global almost optimal error estimates in L 2 - and H 1 -norms are derived for the problem with initial data u 0 ∈ H 0 1 (Ω) ∩ H 2 (Ω). The novelty of our approach is based on managing the interaction of the L1 approximation of the fractional derivative and the time discrete elliptic operator to derive the optimal estimate in H 1 -norm directly. Furthermore, a super convergence result is established when the elliptic operator is self-adjoint with time and space varying coefficients, and as a consequence, an L ∞ error estimate is obtained for 2D problems that too with the initial condition is in H 0 1 (Ω) ∩ H 2 (Ω). All results proved in this paper are valid uniformly as α → 1 − , where α is the order of the Caputo fractional derivative. Numerical experiments are presented to validate our theoretical findings. [ABSTRACT FROM AUTHOR]

We propose a notion of stability for constant k$k$‐mean curvature hypersurfaces in a general Riemannian manifold and we give some applications. When the ambient manifold is a Space Form, our notion coincides with the known one, given by means of the variational problem. [ABSTRACT FROM AUTHOR]

This article sets forth results on the existence, non-existence, uniqueness, and regularities properties, as well as boundary behavior of solutions for singular systems involving mixed local and non-local elliptic operators (see System (S) below). More precisely, we first establish a new weak comparison principle for a singular equation. Afterward, we discuss the non-existence of positive classical solutions, as well as construct suitable ordered pairs of sub-solutions and super-solutions. This allows us to obtain the existence of a pair of positive weak solutions for System (S) by employing Schauder's fixed-point theorem in the associated conical shell. Finally, we adapt a method of Krasnoselsky to establish the uniqueness of such a positive pair of solutions. [ABSTRACT FROM AUTHOR]

In this work, we study the problem of learning a partial differential equation (PDE) from its solution data. PDEs of various types are used to illustrate how much the solution data can reveal the PDE operator depending on the underlying operator and initial data. A data-driven and data-adaptive approach based on local regression and global consistency is proposed for stable PDE identification. Numerical experiments are provided to verify our analysis and demonstrate the performance of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we study the spaces of (d+dc)-harmonic forms and of (d+dΛ)-harmonic forms, a natural generalization of the spaces of Bott–Chern harmonic forms (respectively, symplectic harmonic forms) from complex (respectively, symplectic) manifolds to almost Hermitian manifolds. We apply the same techniques to compact complex surfaces, computing their Bott–Chern and Aeppli numbers and their spaces of (d+dΛ)-harmonic forms. We give several applications to compact quotients of Lie groups by a lattice. [ABSTRACT FROM AUTHOR]

We give a characterization of the singular points of the free boundary ∂{u>0} for viscosity solutions of the nonlinear equation F(D²u)=-χ {u>0}, where F is a fully nonlinear elliptic operator and χ is the characteristic function. This equation models the propagation of a nerve impulse along an axon. We analyze the structure of the free boundary ∂{u>0} near the singular points where u and ∇u vanish simultaneously. Our method uses the stratification approach developed in Dipierro and the author's 2018 paper. In particular, when n=2 we show that near a flat singular free boundary point, ∂{u>0} is a union of four C 1 arcs tangential to a pair of crossing lines. [ABSTRACT FROM AUTHOR]

This paper studies the superlinear convergence of Krylov iterations under the streamline‐diffusion preconditioning operator for convection‐dominated elliptic problems. First, convergence results are given involving the diffusion parameter ε$$ \varepsilon $$. Then the limiting case ε=0$$ \varepsilon =0 $$ is studied on the operator level, and the convergence results are extended to this situation under some conditions, in spite of the lack of compactness of the perturbation operators. An explicit rate is also given. [ABSTRACT FROM AUTHOR]

The paper is concerned with the monotonicity of the principal eigenvalues with respect to diffusion rate for two classes of non-self-adjoint operators: time-periodic parabolic operators and elliptic operators with shear flow. These operators behave similarly to some averaged self-adjoint elliptic operators when the frequency or flow amplitude, referred to as advection rate, is sufficiently large. It was conjectured in [S. Liu and Y. Lou, J. Funct. Anal., 282 (2022), 109338] that the principal eigenvalues are monotone in diffusion rate for large advection, similarly to those self-adjoint elliptic operators. We provide some counterexamples to the conjecture by establishing the high order expansion of the principal eigenvalues for sufficiently large diffusion and advection rates. [ABSTRACT FROM AUTHOR]

The goal of this paper is to explore the asymptotic behaviour of anisotropic problems governed by operators of the pseudo p-Laplacian type when the size of the domain goes to infinity in different directions. [ABSTRACT FROM AUTHOR]

We prove Schauder estimates for elliptic and parabolic problems governed by the degenerate operator ℒ = Δ x + D y ⁢ y + c y ⁢ D y , in the half-space Ω = { (x , y) : x ∈ ℝ N , y > 0 } , under Neumann boundary conditions at y = 0 . [ABSTRACT FROM AUTHOR]

Let. Let and. Consider the formal second-order differential operator in. We show that the closure of is quasi-m-accretive under certain conditions on the coefficients. [ABSTRACT FROM AUTHOR]

The fractional differential equation L^\beta u = f posed on a compact metric graph is considered, where \beta >0 and L = \kappa ^2 - \nabla (a\nabla) is a second-order elliptic operator equipped with certain vertex conditions and sufficiently smooth and positive coefficients \kappa,a. We demonstrate the existence of a unique solution for a general class of vertex conditions and derive the regularity of the solution in the specific case of Kirchhoff vertex conditions. These results are extended to the stochastic setting when f is replaced by Gaussian white noise. For the deterministic and stochastic settings under generalized Kirchhoff vertex conditions, we propose a numerical solution based on a finite element approximation combined with a rational approximation of the fractional power L^{-\beta }. For the resulting approximation, the strong error is analyzed in the deterministic case, and the strong mean squared error as well as the L_2(\Gamma \times \Gamma)-error of the covariance function of the solution are analyzed in the stochastic setting. Explicit rates of convergences are derived for all cases. Numerical experiments for {L = \kappa ^2 - \Delta, \kappa >0} are performed to illustrate the results. [ABSTRACT FROM AUTHOR]

This paper is devoted to dealing with a kind of new Kirchhoff-type problem in RN that involves a general double-phase variable exponent elliptic operator ϕ. Specifically, the operator ϕ has behaviors like |τ|q(x)−2 τ if |τ| is small and like |τ| p(x)−2 τ if |τ| is large, where 1 < p(x) < q(x) < N. By applying some new analytical tricks, we first establish existence results of solutions for this kind of Kirchhoff-double-phase problem based on variational methods and critical point theory. In particular, we also replace the classical Ambrosetti–Rabinowitz type condition with four different superlinear conditions and weaken some of the assumptions in the previous related works. Our results generalize and improve the ones in [Q. H. Zhang, V. D. Rădulescu, J. Math. Pures Appl., 118 (2018), 159–203.] and other related results in the literature. [ABSTRACT FROM AUTHOR]

In this article, we study the numerical corroboration of a variational model governed by a fourth-order elliptic operator that describes the deformation of a linearly elastic flexural shell subjected not to cross a prescribed flat obstacle. The problem under consideration is modelled by means of a set of variational inequalities posed over a non-empty, closed and convex subset of a suitable Sobolev space and is known to admit a unique solution. Qualitative and quantitative numerical experiments corroborating the validity of the model and its asymptotic similarity with Koiter's model are also presented. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'. [ABSTRACT FROM AUTHOR]

We study the representative volume element (RVE) method, which is a method to approximately infer the effective behavior a hom of a stationary random medium. The latter is described by a coefficient field a(x) generated from a given ensemble ⟨ · ⟩ and the corresponding linear elliptic operator - ∇ · a ∇ . In line with the theory of homogenization, the method proceeds by computing d = 3 correctors (d denoting the space dimension). To be numerically tractable, this computation has to be done on a finite domain: the so-called representative volume element, i.e., a large box with, say, periodic boundary conditions. The main message of this article is: Periodize the ensemble instead of its realizations. By this, we mean that it is better to sample from a suitably periodized ensemble than to periodically extend the restriction of a realization a(x) from the whole-space ensemble ⟨ · ⟩ . We make this point by investigating the bias (or systematic error), i.e., the difference between a hom and the expected value of the RVE method, in terms of its scaling w.r.t. the lateral size L of the box. In case of periodizing a(x), we heuristically argue that this error is generically O (L - 1) . In case of a suitable periodization of ⟨ · ⟩ , we rigorously show that it is O (L - d) . In fact, we give a characterization of the leading-order error term for both strategies and argue that even in the isotropic case it is generically non-degenerate. We carry out the rigorous analysis in the convenient setting of ensembles ⟨ · ⟩ of Gaussian type, which allow for a straightforward periodization, passing via the (integrable) covariance function. This setting has also the advantage of making the Price theorem and the Malliavin calculus available for optimal stochastic estimates of correctors. We actually need control of second-order correctors to capture the leading-order error term. This is due to inversion symmetry when applying the two-scale expansion to the Green function. As a bonus, we present a stream-lined strategy to estimate the error in a higher-order two-scale expansion of the Green function. [ABSTRACT FROM AUTHOR]

We consider a second-order elliptic operator with sufficiently smooth variable coefficients in an arbitrary two-dimensional domain with fast oscillating boundary under the assumption that the oscillation amplitude is small. The structure of the oscillations is fairly arbitrary and no periodicity or local periodicity conditions are imposed. The oscillating boundary is divided into two components with the Dirichlet boundary condition posed on one of the components and the Neumann condition on the other. Such mixed boundary conditions are preserved under homogenization; as a result, the functions in the domain of the homogenized operator have weak power-law singularities. Despite these singularities, we succeed to modify the technique from our previous papers appropriately and to prove the uniform resolvent convergence of the perturbed operator to the homogenized operator and estimate the convergence rate. [ABSTRACT FROM AUTHOR]

Given an action of a discrete group on a closed smooth manifold, we consider the class of nonlocal elliptic operators generated by pseudodifferential operators on the manifold and shift operators. The operators in this class are Fredholm under suitable ellipticity conditions. In the present paper, we give an index formula for these operators for the case of the group , where is an arbitrary finite group. To solve this index problem, we explicitly write the contribution to the index by the fixed points of finite-order elements of the group. The index formula is stated in terms of characteristic classes in periodic cyclic cohomology. [ABSTRACT FROM AUTHOR]

In this research, we study certain characteristics of the solution for an equation involving uniformly elliptic nonlocal operator by establishing various maximum principles in bounded and unbounded regions. Additionally, we don't need the conditions that the narrow region principle's bound assumption and decay in the unbounded domain. In order to get the monotonicity of the solution, we use several maximum principles and averaging effects to overcome the challenges that generated by the uniformly elliptic nonlocal operator and fractional‐order variable t$$ t $$. Of course, while investigating the qualitative characteristics of the equation, we also provide some estimates of auxiliary function. For the study of additional nonlocal operators, we believe the method of uniformly elliptic nonlocal operators will also be helpful. [ABSTRACT FROM AUTHOR]