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In the present work, we give an explicit solution of the Dirichlet boundary problem for the Helmholtz equation in a nonconvex angle with periodic boundary data. We present uniqueness and existence theorems in an appropriate functional class and we give an explicit formula for the solution in the form of the Sommerfeld integral. The method of complex characteristics [12] is used. [ABSTRACT FROM AUTHOR]

In this paper, the solvability of a new class of nonlocal boundary value problems for the Poisson equation is studied. Nonlocal conditions are specified in the form of a connection between the values of the unknown function at different points of the boundary. In this case, the boundary operator is determined using matrices of involution-type mappings. Theorems on the existence and uniqueness of solutions to the studied problems are proved. Using Green's functions of the classical Dirichlet and Neumann boundary value problems, Green's functions of the studied problems are constructed and integral representations of solutions to these problems are obtained. [ABSTRACT FROM AUTHOR]

We investigate the concept of effective resistance in connection graphs, expanding its traditional application from undirected graphs. We propose a robust definition of effective resistance in connection graphs by focusing on the duality of Dirichlet-type and Poisson-type problems on connection graphs. Additionally, we delve into random walks, taking into account both node transitions and vector rotations. This approach introduces novel concepts of effective conductance and resistance matrices for connection graphs, capturing mean rotation matrices corresponding to random walk transitions. Thereby, it provides new theoretical insights for network analysis and optimization. [ABSTRACT FROM AUTHOR]

We study difference approximations of an optimal control problem with a boundary observation of the conormal derivative of the state described by the Dirichlet problem for semilinear elliptic equations with controls involved in coefficients of the convective transport operator and in the nonlinear term of the equation. The well-posedness of the optimal control problem is examined. Difference approximations for the optimal control problem are constructed. The convergence of the approximations with respect to the functional and control is analyzed. A regularization of the approximations is constructed. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical approximation of initial‐boundary‐value problems of a three‐parameter family of Bona–Smith systems, derived as a model for the propagation of surface waves under a physical Boussinesq regime. The work proposed here is focused on the corresponding problem with Dirichlet boundary conditions and its approximation in space with spectral methods based on Jacobi polynomials, which are defined from the orthogonality with respect to some weighted L2$L^{2}$ inner product. Well‐posedness of the problem on the corresponding weighted Sobolev spaces is first analyzed and existence and uniqueness of solution, locally in time, are proved. Then, the spectral Galerkin semidiscrete scheme and some detailed comments on its implementation are introduced. The existence of numerical solution and error estimates on those weighted Sobolev spaces are established. Finally, the choice of the time integrator to complete the full discretization takes care of different stability issues that may be relevant when approximating the semidiscrete system. Some numerical experiments illustrate the results. [ABSTRACT FROM AUTHOR]

We consider an eigenvalue problem for the Dirichlet $ (p,q) $ (p , q) -Laplacian. Using truncations and variational tools we prove a global multiplicity result producing at least two positive solutions. [ABSTRACT FROM AUTHOR]

First we study the Dirichlet problem Reω(z)→φ(ζ) as z→ζ, z∈D, ζ∈∂D, with continuous boundary data φ:∂D→R for semi-linear Beltrami equations ωz-μ(z)ωz=σ(z)q(Reω(z)). We assume here that D is an arbitrary bounded domain of the complex plane C, which is either simply connected or has no boundary components degenerated to a single point, and that the equations are locally uniform elliptic with possible singularities at the boundary. For σ∈Lp(D), p>2, with compact support, and continuous q:R→C, q(t)/t→0 as t→∞, we establish a series of effective criteria for existence of solutions of the Dirichlet problem in terms of BMO, FMO, Calderon-Zygmund, Lehto and Orlicz integral means. We also establish representation and regularity of these solutions. Then, we prove existence, representation and regularity of weak solutions of the Dirichlet problem u(z)→φ(ζ) as z→ζ, z∈D,ζ∈∂D, to semi-linear Poisson type equations div[A(z)∇u(z)]=g(z)Q(u(z)) for g∈Lp(D), p>1, with compact support, and continuous Q:R→R, Q(t)/t→0 as t→∞. We also assume here conditions on the matrix coefficients A(z) guaranteing locally uniform ellipticity of these equations. Finally, we give examples of possible applications of the obtained results to various semi-linear equations of the mathematical physics in anisotropic and inhomogeneous media. [ABSTRACT FROM AUTHOR]

In this paper, we present a comprehensive study of second-order differential operators on a star-graph geometric graph considering a star graph with three edges and a common vertex. We investigate the Dirichlet problem for a Sturm-Liouville operator defined on this network-type manifold. The Sturm-Liouville problem is formulated as a system of ordinary differential equations on the individual edges, subject to the boundary conditions at the common vertex. We assume that the condition holds, ensure the non-degeneracy of the boundary conditions by using a synthetic approach. We fully describe and solve the Dirichlet problem for the given second-order differential operator on the star graph. The key results include the characterization of the spectral parameter λ, the construction of the matrix A composed of the boundary condition coefficients, and the analysis of the minors of A. The findings of this work contribute to the understanding of second-order differential operators on network-type manifolds and provide a framework for addressing similar problems on more complex geometric graphs. The insights gained from this study have potential applications in various fields, such as: quantum mechanics, control theory, and network analysis. [ABSTRACT FROM AUTHOR]

The RSS approach is a useful method of sampling that reduces the cost and improves the representativeness of the population. It provides more efficient estimators than the competitors based on SRS. However, using RSS could be a difficult task to observe all the ranks. Thus, using only the extreme ranks eases the task and reduces the error in ranking. Samawi et al. (1996) proposed the method of Extreme Ranked Set Sampling (ERSS) to reduce errors in ranking and showed that the method gives an unbiased estimate of the population mean in the case of symmetric populations and it provides a more efficient estimator than SRS. However, the estimator of this method is biased when the distribution is skewed. Many researchers have considered ERSS, investigated several estimators, and studied their properties. In this paper, we adopt the ERSS technique when the samples are drawn from the Rician distribution. Several estimators have been studied, including arithmetic mean, geometric mean, harmonic mean, quadratic mean, median, variance, mean deviation, skewness, and kurtosis. Computer simulations were used to check the properties of these estimators and compared with the corresponding estimators using SRS. Some estimators based on ERSS are more efficient than the corresponding estimators from SRS, but some others are not. [ABSTRACT FROM AUTHOR]

This paper concerns a stationary prey-predator model with densitydependent diffusion and hunting cooperation under homogeneous Dirichlet boundary conditions. Based on the spectral analysis, the asymptotic stability of trivial and semi-trivial solutions is obtained. Moreover, the sufficient conditions for the existence of positive solutions are established by using degree theory in cones. Our analytical results suggest that density-dependent diffusion and hunting cooperation obviously influence on the positive solutions. [ABSTRACT FROM AUTHOR]

In this paper, we employ the finite element method based on non-uniform rational B-splines function approximation to solve the nonlinear Brinkman-Forcheimer-Darcy equation in a simply connected and bounded Lipschitz domain. We provide both theoretical and numerical studies of the Dirichlet boundary problem. Utilizing a stream function formulation, we demonstrate the well-posedness of the weak form. Furthermore, we approximate the velocity and pressure fields by linearizing the nonlinear terms, resulting in an algebraic system. This Non-uniform rational B-splines method is more effective in terms of the exact representation of the geometry and the good approximation of the solution compared to the virtual element method. To validate the effectiveness of the non-uniform rational B-splines Finite Element Method, we conduct numerical simulations of fluid flow in porous media. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Dirichlet problem in a bounded open set Ω ⊂ R n ( n ⩾ 2 ) for a class of second-order nonlinear elliptic equations with right-hand side f in L 1 (Ω) . We study the summability of entropy and weak solutions of this problem under the stronger assumption that f G (| f |) ∈ L 1 (Ω) , where G is a nonnegative increasing continuous function on [ 0 , + ∞) . We show how the summability of the solutions depends on the function G. Our conditions on G imply that L 1 + ε (Ω) ⊂ K G ⊂ L 1 (Ω) for every ε > 0 , where K G is the set of all measurable functions v on Ω such that v G (| v |) ∈ L 1 (Ω) . [ABSTRACT FROM AUTHOR]

This paper investigates the existence of weak solutions for (p, q)-Laplacian problems where p and q depend on the unknown solution. We focus on the case where p and q are local quantities. By means of a singular perturbation technique, we prove the existence of weak solutions for certain Dirichlet problem. [ABSTRACT FROM AUTHOR]

Let (X, d, μ) be a metric measure space satisfying the doubling condition and an L2-Poincaré inequality. Consider the nonnegative operator L generalized by a Dirichlet form on X. We will show that a solution u to (− ∂ t 2 + L) u = 0 on X × ℝ+ satisfies an α-Carleson condition if and only if u can be represented as the Poisson integral of the operator L with the trace in the generalized Morrey space L2,α(X), where α is a nonnegative function defined on a class of balls in X. This result extends the analogous characterization founded by R. Jiang, J. Xiao, D. Yang (2016) from the classical Morrey space on Euclidean space to the generalized Morrey space on the metric measure space. [ABSTRACT FROM AUTHOR]

In this article, we consider the Dirichlet problem for nonlinear higher-order equations in upper half plane. Firstly we introduce the solutions of inhomogeneous polyanalytic equation in upper half plane. Then we investigate the properties of relevant integral operators. Lastly we transform the Dirichlet problem for nonlinear higher-order equations in upper half plane into the system of integro-differential equations and we obtain the existence of unique solution using Banach fixed point theorem. [ABSTRACT FROM AUTHOR]

We consider approximating the solution of the Helmholtz exterior Dirichlet problem for a nontrapping obstacle, with boundary data coming from plane-wave incidence, by the solution of the corresponding boundary value problem where the exterior domain is truncated and a local absorbing boundary condition coming from a Padé approximation (of arbitrary order) of the Dirichlet-to-Neumann map is imposed on the artificial boundary (recall that the simplest such boundary condition is the impedance boundary condition). We prove upper- and lower-bounds on the relative error incurred by this approximation, both in the whole domain and in a fixed neighbourhood of the obstacle (i.e. away from the artificial boundary). Our bounds are valid for arbitrarily-high frequency, with the artificial boundary fixed, and show that the relative error is bounded away from zero, independent of the frequency, and regardless of the geometry of the artificial boundary. [ABSTRACT FROM AUTHOR]

We investigate the qualitative properties of weak solutions to the boundary value problems for fourth-order linear hyperbolic equations with constant coefficients in a plane bounded domain convex with respect to characteristics. Our main scope is to prove some analog of the maximum principle, solvability, uniqueness and regularity results for weak solutions of initial and boundary value problems in the space L2. The main novelty of this paper is to establish some analog of the maximum principle for fourth-order hyperbolic equations. This question is very important due to natural physical interpretation and helps to establish the qualitative properties for solutions (uniqueness and existence results for weak solutions). The challenge to prove the maximum principle for weak solutions remains more complicated and at that time becomes more interesting in the case of fourth-order hyperbolic equations, especially, in the case of non-classical boundary value problems with data of weak regularity. Unlike second-order equations, qualitative analysis of solutions to fourth-order equations is not a trivial problem, since not only a solution is involved in boundary or initial conditions, but also its high- order derivatives. Other difficulty concerns the concept of weak solution of the boundary value problems with L2 – data. Such solutions do not have usual traces, thus, we have to use a special notion for traces to poss correctly the boundary value problems. This notion is traces associated with operator L or L-traces. We also derive an interesting interpretation (as periodicity of characteristic billiard or the John's mapping) of the Fredholm's property violation. Finally, we discuss some potential challenges in applying the results and proposed methods.

We consider propagation of two-dimensional waves on the infinite hexagonal (honeycomb) lattice. Namely, we study the discrete Helmholtz equation in hexagonal lattices without and with a boundary. It is shown that for some configurations these problems can be equivalently reduced to similar problems for the triangular lattice. Based on this fact, new results are obtained for the existence and uniqueness of the solution in the case of the real wave number k ∈ ( 0 , 6 ) ∖ { 2 , 3 , 2 } {k\in(0,\sqrt{6})\setminus\{\sqrt{2},\sqrt{3},2\}} for the non-homogeneous Helmholtz equation in hexagonal lattices with no boundaries and the real wave number k ∈ ( 0 , 2 ) ∪ ( 2 , 6 ) {k\in(0,\sqrt{2})\cup(2,\sqrt{6})} for the exterior Dirichlet problem. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic dynamics for a class of semilinear parabolic problems with Dirichlet boundary conditions in domains with a small hole of size proportional to a positive parameter ε. In other words, we prove that the family of attractors behaves continuously as ε → 0. We will also provide the convergence rates in terms of the parameter. [ABSTRACT FROM AUTHOR]

In this paper, we study a noncoercive nonlinear elliptic operator with a drift term in an unbounded domain. The singular first-order term grows like | E (x) | | ∇ u | , where E (x) is a vector field belonging to a suitable Morrey-type space. Our operator arises as a stationary equation of diffusion–advection problems. We prove existence, regularity, and uniqueness theorems for a Dirichlet problem. To obtain our main results, we use the weak maximum principle and the same a priori estimates. [ABSTRACT FROM AUTHOR]

The paper considers a second-order quasilinear elliptic equation with an integrable right-hand side. Restrictions on the structure of the equation are stated in terms of the generalized -function. Unlike the author's previous papers, there is no sign condition on the lower-order term of the equation. The existence of a renormalized solution of the Dirichlet problem for this equation is proved in nonreflexive Musielak–Orlicz–Sobolev spaces in an arbitrary unbounded strictly Lipschitz domain. [ABSTRACT FROM AUTHOR]

There is a vast theory of Chebyshev and residual polynomials and their asymptotic behavior. The former ones maximize the leading coefficient and the latter ones maximize the point evaluation with respect to an L ∞ norm. We study Chebyshev and residual extremal problems for rational functions with real poles with respect to subsets of R ¯ . We prove root asymptotics under fairly general assumptions on the sequence of poles. Moreover, we prove Szegő–Widom asymptotics for sets which are regular for the Dirichlet problem and obey the Parreau–Widom and DCT conditions. [ABSTRACT FROM AUTHOR]

We consider the difference maximum principle with input data of variable sign and its application to the investigation of the monotonicity and convergence of finite-difference schemes (FDSs). Namely, we consider the Dirichlet initial-boundary-value problem for multidimensional quasilinear parabolic equations with unbounded nonlinearity. Unconditionally monotone linearized finite-difference schemes of the second-order of accuracy are constructed on uniform grids. A two-sided estimate for the grid solution, which is completely consistent with similar estimates for the exact solution, is obtained. These estimates are used to prove the convergence of FDSs in the grid L2-norm. We also present a study aimed at constructing second-order monotone difference schemes for the parabolic convection-diffusion equation with boundary conditions of the third kind and unlimited nonlinearity without using the initial differential equation on the domain boundaries. The goal is a combination of the assumption of existence and uniqueness of a smooth solution and the regularization principle. In this case, the boundary conditions are directly approximated on a two-point stencil of the second order. [ABSTRACT FROM AUTHOR]

In this paper we consider the homogenization of the evolution problem associated with a jump process that involves three different smooth kernels that govern the jumps to/from different parts of the domain. We assume that the spacial domain is divided into a sequence of two subdomains A n ∪ B n and we have three different smooth kernels, one that controls the jumps from A n to A n , a second one that controls the jumps from B n to B n and the third one that governs the interactions between A n and B n . Assuming that χ A n (x) → X (x) weakly in L ∞ (and then χ B n (x) → 1 - X (x) weakly in L ∞ ) as n → ∞ and that the initial condition is given by a density u 0 in L 2 we show that there is an homogenized limit system in which the three kernels and the limit function X appear. When the initial condition is a delta at one point, δ x ¯ (this corresponds to the process that starts at x ¯ ) we show that there is convergence along subsequences such that x ¯ ∈ A n j or x ¯ ∈ B n j for every n j large enough. We also provide a probabilistic interpretation of this evolution equation in terms of a stochastic process that describes the movement of a particle that jumps in Ω according to the three different kernels and show that the underlying process converges in distribution to a limit process associated with the limit equation. We focus our analysis in Neumann type boundary conditions and briefly describe at the end how to deal with Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate variational problems in R 2 with the Sobolev norm constraints and with the Dirichlet norm constraints. We focus on property of maximizers of the variational problems. Concerning variational problems with the Sobolev norm constraints, we prove that maximizers are ground state solutions of corresponding elliptic equations, while we exhibit an example of a ground state solution which is not a maximizer of corresponding variational problems. On the other hand, we show that maximizers of maximization problems with the Dirichlet norm constraints and ground state solutions of corresponding elliptic equations are the same functions, up to scaling, under suitable setting. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of weak solutions for a class of degenerate elliptic Dirichlet problems with critical nonlinearity and a logarithmic perturbation, i.e. 0.2 { - (Δ x u + (α + 1) 2 | x | 2 α Δ y u) = u Q + 2 Q - 2 + λ u log u 2 , u = 0 on ∂ Ω , where (x , y) ∈ Ω ⊂ R N = R m × R n with m ≥ 1 , n ≥ 0 , Ω ∩ { x = 0 } ≠ ∅ is a bounded domain, the parameter α ≥ 0 and Q = m + n (α + 1) denotes the "homogeneous dimension" of R N . When λ = 0 , we know that from [23] the problem (0.2) has a Pohožaev-type non-existence result. Then for λ ∈ R \ { 0 } , we establish the existences of non-negative ground state weak solutions and non-trivial weak solutions subject to certain conditions. [ABSTRACT FROM AUTHOR]

In this paper, we study a Riemann-Hilbert problem related to inhomogeneous complex partial differential operators of higher order on the unit disk. Applying the Cauchy-Pompeiu formula, we find out the solvable conditions and obtain the representation of the solutions. Then, we investigate the boundary value problems for bi-polyanalytic functions with the Dirichlet and Riemann-Hilbert boundary conditions, obtain the specific solution and the solvable conditions, and extend the conclusion to the corresponding higher-order problems. Therefore, we obtain the solution to the half-Neumann problem of higher order for bi-polyanalytic functions. [ABSTRACT FROM AUTHOR]

In this paper, we study the Dirichlet problem of translating mean curvature equations over a domain via topological restrictions. Its main difficulty is the non-existence of a C0 a priori estimate for their classical solutions except few cases. Inspired by the work of Miranda and Giusti, we define a generalized solution of these Dirichlet problems and establish its general existence. We propose a nonclosed- minimal (NCM) condition on the underlying domain. When the domain is mean convex and NCM, the generalized solution with continuous boundary data is the classical smooth solution. Moreover, the NCM assumption cannot be removed by a hemisphere example. [ABSTRACT FROM AUTHOR]

In the unit disc, we consider the Dirichlet problem for second order elliptic system with only higher order constant coefficients. We establish the unique solvability of this problem under the assumption that the problem is Fredholm, and obtain an explicit formula for the solution. [ABSTRACT FROM AUTHOR]

We consider a nonlinear parabolic partial differential equation in the case where the unknown function depends on two spatial variables and time, which is a generalization of the well-known Kuramoto–Sivashinsky equation. We consider homogeneous Dirichlet boundary-value problems for this equation. We examine local bifurcations when spatially homogeneous equilibrium states change stability. We show that post-critical bifurcations are realized in the boundary-value problems considered. We obtain asymptotic formulas for solutions and examine the stability of spatially inhomogeneous solutions. [ABSTRACT FROM AUTHOR]

In the current work, we study the geometry and topology of warped product Legendrian submanifolds in Kenmotsu space forms F 2 n + 1 (ϵ) and derive the first Chen inequality, including extrinsic invariants such as the mean curvature and the length of the warping functions. Additionally, sectional curvature and the δ-invariant are intrinsic invariants related to this inequality. An integral bound is also given in terms of the gradient Ricci curvature for the Bochner operator formula of compact warped product submanifolds. Our primary technique is applying geometry to number structures and solving problems such as problems with Dirichlet eigenvalues. [ABSTRACT FROM AUTHOR]

We consider a semilinear Dirichlet problem with a reaction which exhibits an asymmetric behaviour as $ x \to \pm \infty $ x → ± ∞ (it is superlinear as $ x \to +\infty $ x → + ∞ and resonant as $ x \to -\infty $ x → − ∞). Using variational tools from the critical point theory, together with truncation and comparison techniques and critical groups, we prove two multiplicity theorems producing two and three nontrivial solutions. Also we show that the problem can not have negative solutions. [ABSTRACT FROM AUTHOR]

We treat the exterior Dirichlet problem for a class of fully nonlinear elliptic equations of the form f (λ (D 2 u)) = g (x) with prescribed asymptotic behavior at infinity. The equations of this type had been studied extensively by Caffarelli et al. [The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985) 261–301], Trudinger [On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995) 151–164] and many others, and there had been significant discussions on the solvability of the classical Dirichlet problem via the continuity method, under the assumption that f is a concave function. In this paper, based on Perron's method, we establish an exterior existence and uniqueness result for viscosity solutions of the equations, by assuming f to satisfy certain structure conditions as in [L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985) 261–301; N. S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995) 151–164] but without requiring the concavity of f. The equations in our setting may embrace the well-known Monge–Ampère equations, Hessian equations and Hessian quotient equations as special cases. [ABSTRACT FROM AUTHOR]

We consider a nonlinear Dirichlet problem driven by the (p (z) , q) -Laplacian and with a logistic reaction of the equidiffusive type. Under a nonlinearity condition on a quotient map, we show existence and uniqueness of positive solutions and the result is global in parameter λ. If the monotonicity condition on the quotient map is not true, we can no longer guarantee uniqueness, but we can show the existence of a minimal solution u λ * and establish the monotonicity of the map λ ⟼ u λ * and its asymptotic behaviour as the parameter λ decreases to the critical value λ ^ 1 (q) > 0 (the principal eigenvalue of (− Δ q , W 0 1 , q (Ω)) ). [ABSTRACT FROM AUTHOR]

We prove Besov boundary regularity for solutions of the homogeneous Dirichlet problem for fractional-order quasi-linear operators with variable coefficients on Lipschitz domains Ω of Rd. Our estimates are consistent with the boundary behavior of solutions on smooth domains and apply to fractional p-Laplacians and operators with finite horizon. The proof exploits the underlying variational structure and uses a new and flexible local translation operator. We further apply these regularity estimates to derive novel error estimates for finite element approximations of fractional p-Laplacians and present several simulations that reveal the boundary behavior of solutions. [ABSTRACT FROM AUTHOR]

Unfitted finite element methods (FEM) have attractive merits for problems with evolving or geometrically complex boundaries. Conventional unfitted FEMs incorporate penalty terms, parameters, or Lagrange multipliers to impose the Dirichlet boundary condition weakly. This to some extent increases computational complexity in implementation. In this article, we propose an unfitted generalized FEM (GFEM) for the Dirichlet problem, which is free from any penalty or stabilization. This is achieved by means of partition of unity frameworks of GFEM and designing a set of new enrichments for the Dirichlet boundary. The enrichments are divided into two groups: the one is used to impose the Dirichlet boundary condition strongly, and the other one serves as energy space of variational formulations. The shape functions in energy space vanish at the boundary so that standard variational formulae like those in the conventional fitted FEM can be applied, and thus the penalty and stabilization are not needed. The optimal convergence rate in the energy norm is proven rigorously. Numerical experiments and comparisons with other methods are executed to verify the theoretical result and effectiveness of the algorithm. The conditioning of new method is numerically shown to be of same order as that of the standard FEM. [ABSTRACT FROM AUTHOR]

We improve unconditional estimates on Δk(x)$\Delta _k(x)$, the remainder term of the generalised divisor function, for large k$k$. In particular, we show that Δk(x)≪x1−1.889k−2/3$\Delta _k(x) \ll x^{1 - 1.889k^{-2/3}}$ for all sufficiently large fixed k$k$. [ABSTRACT FROM AUTHOR]

This paper is devoted to the mathematical analysis of the contact capabilities of the fluid-structure interaction (FSI) model with seepage reported in [Comput. Methods Appl. Mech., 392:114637, 2022]. In the case of a rigid disk moving over a fixed horizontal plane, we show that this model encompasses contact and hence removes the non collision paradox of traditional FSI models which rely on Dirichlet or Dirichlet/Navier boundary conditions. Numerical evidence on the theoretical results is also provided. [ABSTRACT FROM AUTHOR]

For surfaces of revolution we prove the existence of infinitely many sign-changing rotationally symmetric solutions to a wide class of $ p $-Laplace-Beltrami equations with polynomial like nonlinearities. We combine the methods in [10] where the semilinear case $ (p = 2) $ with positive weight was studied with those in [5] and [6] where radial solutions to a $ p $-Laplacian Dirichlet problem in a ball was considered. Our equations lead to ordinary differential equations with two singularities and a sign changing weight function which leads to an initial value problem that may blow up in the region of definition of the equation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Dirichlet problem of logarithmic Laplacian equations and systems on unbounded regions. We first establish new narrow region principles for the logarithmic Laplacian equation and system in coercive epigraphs separately. Then by using the direct method of moving planes, we obtain a monotonicity result for the solutions of logarithmic Laplacian equations and a Liouville-type theorem for solutions of logarithmic Laplacian systems in coercive epigraphs. [ABSTRACT FROM AUTHOR]

In this paper, we deduce a new formula that explicitly expresses the cylindrical Poisson kernel of any degree and order using cylindrical Poisson polynomials. We then demonstrate how this formula can be applied to solve the Dirichlet boundary value problems for cylindrical Laplace equations. Furthermore, we examine and discuss the behaviors of this solution approach at infinity. [ABSTRACT FROM AUTHOR]

We consider local solutions u of nonlinear elliptic systems of the type div A (x , D u) = div F in Ω ⊂ R n , where u : Ω → R N is in a weighted W loc 1 , p space, with p ≥ 2 , F is in a weighted W loc 1 , 2 space and x → A (x , ξ) has growth coefficients in the space of functions with bounded mean oscillation. We prove higher differentiability of u in the sense that the nonlinear expression of its gradient V μ (D u) : = (μ 2 + | D u | 2 ) p - 2 4 D u , with 0 < μ ≤ 1 , is weakly differentiable with D (V μ (D u)) in a weighted L loc 2 space. Moreover we derive some local Calderón–Zygmund estimates when the source term is not necessarily differentiable. Global estimates for a suitable Dirichlet problem are also available. [ABSTRACT FROM AUTHOR]

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We investigate error bounds for numerical solutions of divergence structure linear elliptic partial differential equations (PDEs) on compact manifolds without boundary. Our focus is on a class of monotone finite difference approximations, which provide a strong form of stability that guarantees the existence of a bounded solution. In many settings including the Dirichlet problem, it is easy to show that the resulting solution error is proportional to the formal consistency error of the scheme. We make the surprising observation that this need not be true for PDEs posed on compact manifolds without boundary. We propose a particular class of approximation schemes built around an underlying monotone scheme with consistency error |$O(h^{\alpha })$|⁠. By carefully constructing barrier functions, we prove that the solution error is bounded by |$O(h^{\alpha /(d+1)})$| in dimension |$d$|⁠. We also provide a specific example where this predicted convergence rate is observed numerically. Using these error bounds, we further design a family of provably convergent approximations to the solution gradient. [ABSTRACT FROM AUTHOR]

Discrete duality finite volume (DDFV) schemes are known for their ability to approximate nonlinear and linear anisotropic diffusion operators on general meshes, but they possess several drawbacks. The most important drawback of DDFV is the simultaneous use of the cell and the node unknowns. We propose a discretization approach that incorporates DDFV ideas and the associated analysis techniques, but allows for a rapid elimination of the cell unknowns. Further, unlike the DDFV scheme, the new 'Nodal Discrete Duality' (NDD) scheme does not require specific adaptation in presence of discontinuities of the diffusion tensor along cell boundaries. We describe in detail the 2D NDD framework and its two 3D variants, focusing on the consistency properties of the discrete gradient and discrete divergence operators and on the key structural property of discrete duality. For the 2D scheme, convergence analysis is carried out and a series of numerical tests are provided for a large family of nonlinear anisotropic elliptic problems with zero Dirichlet boundary condition. [ABSTRACT FROM AUTHOR]