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In this paper, we study degenerate or singular elliptic equations in divergence form - div (x n α A ∇ u) = div (x n α g) in B 1 ∩ { x n > 0 }. When α > - 1 , we establish boundary Schauder type estimates under the conormal boundary condition on the flat boundary, provided that the coefficients satisfy Dini mean oscillation (DMO) type conditions. Additionally, as an application, we derive higher-order boundary Harnack principles for uniformly elliptic equations in divergence form with DMO coefficients. [ABSTRACT FROM AUTHOR]

We investigate Bernoulli free boundary problems prescribing infinite jump conditions. The mathematical set-up leads to the analysis of non-differentiable minimization problems of the form ∫ ∇ u · (A (x) ∇ u) + φ (x) 1 { u > 0 } d x → min , where A(x) is an elliptic matrix with bounded, measurable coefficients and φ is not necessarily locally bounded. We prove universal Hölder continuity of minimizers for the one- and two-phase problems. Sharp regularity estimates along the free boundary are also obtained. Furthermore, we perform a thorough analysis of the geometry of the free boundary around a point ξ of infinite jump, ξ ∈ φ - 1 (∞) . We show that it is determined by the blow-up rate of φ near ξ and we obtain an analytical description of such cusp geometries. [ABSTRACT FROM AUTHOR]

Incompressible Euler flows in narrow domains, in which the horizontal length scale is much larger than other scales, play an important role in many different applications, and their leading-order behavior can be described by the hydrostatic Euler equations. In this paper, we show that steady solutions of the hydrostatic Euler equations in an infinite strip strictly away from stagnation must be shear flows. Furthermore, we prove the existence, uniqueness, and asymptotic behavior of global steady solutions to the hydrostatic Euler equations in general nozzles. In terms of stream function formulation, the hydrostatic Euler equations can be written as a degenerate elliptic equation, for which the Liouville type theorem in a strip is a consequence of the analysis for the second order ordinary differential equation (ODE). The analysis on the associated ODE also helps determine the far field behavior of solutions in general nozzles, which plays an important role in guaranteeing the equivalence of stream function formulation. One of the key ingredients for the analysis on flows in a general nozzle is a new transformation, which combines a change of variable and an Euler–Lagrange transformation. With the aid of this new transformation, the solutions in the new coordinates enjoy explicit representations so that the regularity with respect to the horizontal variable can be gained in a clear way. [ABSTRACT FROM AUTHOR]

We study Lipschitz critical points of the energy ∫ Ω g (det D u) d x in two dimensions, where g is a strictly convex function. We prove that the Jacobian of any Lipschitz critical point is constant, and that the Jacobians of sequences of approximately critical points converge strongly. The latter result answers, in particular, an open problem posed by Kirchheim, Müller and Šverák in 2003. [ABSTRACT FROM AUTHOR]

For the problem ∂ t k u - ∑ | α | = m ∂ α a α (x , t , u) ≥ f (| u |) in R + n + 1 = R n × (0 , ∞) , u (x , 0) = u 0 (x) , ∂ t u (x , 0) = u 1 (x) , … , ∂ t k - 1 u (x , 0) = u k - 1 (x) ≥ 0 , where u i ∈ L 1 , l o c (R n) and a α are Caratheodory functions such that | a α (x , t , ζ) | ≤ A | ζ | p , A , p = c o n s t > 0 , for almost all (x , t) ∈ R + n + 1 and for all ζ ∈ R , we obtain exact conditions on the function f guaranteeing that any global weak solution is identically zero. [ABSTRACT FROM AUTHOR]

In this paper we analyze the existence of solution to the homogeneous Dirichlet for a nonlinear elliptic singular problem, $$\begin{align*} \begin{cases} -\Delta u=\dfrac{|\nabla u|^q}{|u|^\alpha}+\lambda f & {\rm in}\ \Omega, \\ u=0 & {\rm on}\ \partial\Omega,\\ \end{cases} \end{align*} $$ { − Δ u = | ∇ u | q | u | α + λf in Ω , u = 0 on ∂Ω , where $ N\ge 2 $ N ≥ 2 , $ \Omega \subset \mathbb {R}^N $ Ω ⊂ R N is a bounded regular domain, q>2, $ \alpha α < q 2 , $ \lambda >0 $ λ > 0 and f is a nonnegative function belonging to a suitable Lebesgue space. Under additional hypotheses on the data, we are also able to show the uniqueness of the 'good' solution. [ABSTRACT FROM AUTHOR]

In this note, we classify solutions to a class of Monge–Ampère equations whose right hand side may be degenerate or singular in the half space. Solutions to these equations are special solutions to a class of fourth order equations, including the affine maximal hypersurface equation, in the half space. Both the Dirichlet boundary value and Neumann boundary value cases are considered. [ABSTRACT FROM AUTHOR]

The present paper is concerned by the study of a nonlinear elliptic equation which contains a Hardy potential, lower order term, singular term and L m (·) data. Our approach is based on approximating the initial problem with a non-singular problem that is well-posed. We then establish the necessary estimates to pass to the limit. [ABSTRACT FROM AUTHOR]

Let (X, G) be a G-system, where G is an infinite countable discrete amenable group and X is a compact metric space with a metric d. In this paper, we study the topological pressure of intricacy and average sample complexity for amenable group actions. We show that the topological pressure of strong sub-additive potential is equal to the pressure of intricacy and average sample complexity as taking supremum over open covers of X. We establish the definition of the pressure of intricacy and average sample complexity by using spanning sets and separated sets. In the last part, we show that the variational principle for pressure of intricacy and average sample complexity. [ABSTRACT FROM AUTHOR]

We consider the numerical approximation of variational problems with orthotropic growth, that is those where the integrand depends strongly on the coordinate directions with possibly different growth in each direction. Under realistic regularity assumptions we derive optimal error estimates. These estimates depend on the existence of an orthotropically stable interpolation operator. Over certain meshes we construct an orthotropically stable interpolant that is also a projection. Numerical experiments illustrate and explore the limits of our theory. [ABSTRACT FROM AUTHOR]

The aim of the paper is to study a quasilinear Dirichlet problem driven by a p-Laplacian type operator with unbounded coefficient and exhibiting a convection term composed with an intrinsic operator. The existence of a bounded weak solution is established. Applications address equations with convolution, truncation, and inverse of p-Laplacian. [ABSTRACT FROM AUTHOR]

Using Mountain Pass Theorem, we consider the existence of weak solutions of weighted Robin problem involving p. -biharmonic operator a x Δ p x 2 u = λ b (x) u q (x) - 2 u , i n Ω a (x) Δ u p (x) - 2 ∂ u ∂ υ + β (x) u p (x) - 2 u = 0 , o n ∂ Ω under some conditions in the space W a , b 2 , p (.) Ω. [ABSTRACT FROM AUTHOR]

In this article, we mainly establish the local extra fractional differentiability (Besov regularity) of weak solutions for the following divergence nonlinear elliptic equations of pp-Laplacian type: divA(Du,x)=divF,\hspace{0.1em}\text{div}\hspace{0.1em}A(Du,x)=\hspace{0.1em}\text{div}\hspace{0.1em}{\bf{F}}, where AA is a Carathéodory function with p−1p-1 growth for 1

We prove sharp upper and lower estimates for the parabolic kernel of the singular elliptic operator L = Tr A D 2 + v , ∇ y , in the half-space R + N + 1 = { (x , y) : x ∈ R N , y > 0 } under Neumann or oblique derivative boundary conditions at y = 0 . [ABSTRACT FROM AUTHOR]

This article is devoted to study a class of new p(x)p\left(x)-Kirchhoff equation. By means of perturbation technique, variational method, and the method invariant sets for the descending flow, the existence and multiplicity of solutions to this problem are obtained.

In the present study, we investigate the existence of the normalized solutions to Sobolev critical fractional Schrödinger equation: (−Δ)su+λu=f(u)+∣u∣2s*−2u,inRN,(Pm)∫RN∣u∣2dx=m2,\hspace{14em}\left\{\begin{array}{ll}{\left(-\Delta )}^{s}u+\lambda u=f\left(u)+{| u| }^{{2}_{s}^{* }-2}u,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{12em}\left({P}_{m})\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}{\rm{d}}x={m}^{2},\hspace{1.0em}\end{array}\right. where 00m\gt 0, 2s*≔2NN−2s{2}_{s}^{* }:= \frac{2N}{N-2s}, λ\lambda is an unknown parameter that will appear as a Lagrange multiplier, and ff is a mass supercritical and Sobolev subcritical nonlinearity. Under fairly general assumptions about ff, with the aid of the Pohozaev manifold and concentration-compactness principle, we obtain a couple of the normalized solution to (Pm)\left({P}_{m}). We mainly extend the results of Appolloni and Secchi (Normalized solutions for the fractional NLS with mass supercritical nonlinearity, J. Differential Equations 286 (2021), 248–283) concerning the above problem from Sobolev subcritical setting to Sobolev critical setting, and also extend the results of Jeanjean and Lu (A mass supercritical problem revisited, Calc. Var. 59 (2020), 174) from classical Schrödinger equation to fractional Schrödinger equation involving Sobolev critical growth. More importantly, our result settles an open problem raised by Soave (Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case, J. Funct. Anal. 279 (2020), 108610), when s=1s=1.

The Dirichlet problem of elliptic equation from two phase problems, b (u) − div (| ∇ u) | p (x) − 2 ∇ u + μ (x) | ∇ u | q (x) − 2 ∇ u) = f (x) in Ω, u = 0 on ∂Ω is considered, where Ω is a bounded domain with a smooth boundary in R N , b is a continuous and nondecreasing function. By the theory of the weighted variable Sobolev space, the existence and uniqueness of an entropy solution for L 1 -data f are proved. [ABSTRACT FROM AUTHOR]

In this article, we study some anisotropic singular perturbations for a class of linear elliptic problems. A uniform estimates for conforming $ Q_1 $ Q 1 finite element method are derived, and some other results of convergence and regularity for the continuous problem are proved. The arguments are based on the tensorization trick and standard elliptic regularity theory tools. [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]

Suppose Ω 0 , Ω 1 are two bounded strongly C -convex domains in C n , with n ≥ 2 and Ω 1 ⊃ Ω 0 ¯ . Let R = Ω 1 \ Ω 0 ¯ . We call R a C -convex ring. We will show that for a solution Φ to the homogenous complex Monge–Ampère equation in R , with Φ = 1 on ∂ Ω 1 and Φ = 0 on ∂ Ω 0 , - 1 ∂ ∂ ¯ Φ has rank n - 1 and the level sets of Φ are strongly C -convex. [ABSTRACT FROM AUTHOR]

Within this article, the maximal function method is used to establish the Calderón-Zygmund estimates for the weak solutions of a class of non-uniformly elliptic equations - div A (x , D u) = - div F (x , f) i n R n , where A (x , D u) ≈ | D u | p 1 - 2 + μ (x) | D u | p 2 - 2 , F (x , f) ≈ | f | p 1 - 2 + μ (x) | f | p 2 - 2 and 1 < p 1 < p 2 , 0 ≤ μ (·) ∈ C 0 , α (R n) , α ∈ (0 , 1 ] . The aforementioned problems arise as Euler-Lagrange equations for variational functionals that were originally presented and studied within the context of Homogenization and the Lavrentiev phenomenon by Marcellini (Arch Ration Mech Anal 105:267–284, 1989. ) and Zhikov (Izv Akad Nauk SSSR Ser Mat 29:33–66, 1987. ). They are distinctive in that they exhibit that the growth and ellipticity change between two distinct types of polynomial depending on the position. This feature is characteristic of strongly anisotropic materials. The contribution of this paper is closely tied to the significant advancements made by Colombo and Mingione (J Funct Anal 270:1416–1478, 2016. ) in the qualitative analysis of double phase problems, as well as the related techniques used by Zhang et al. (Ann Polon Math, 114:45–65, 2015. ). [ABSTRACT FROM AUTHOR]

We study a conormal boundary value problem for a class of quasilinear elliptic equations in bounded domain Ω whose coefficients can be degenerate or singular of the type dist (x , ∂ Ω) α , where ∂ Ω is the boundary of Ω and α ∈ (- 1 , ∞) is a given number. We establish weighted Sobolev type estimates for weak solutions under a smallness assumption on the weighted mean oscillations of the coefficients in small balls. Our approach relies on a perturbative method and several new Lipschitz estimates for weak solutions to a class of singular-degenerate quasilinear equations. [ABSTRACT FROM AUTHOR]

Let X = { X 0 , … , X m } be a family of smooth vector fields on an open set Ω ⊆ R N . Motivated by applications to the PDE theory of Hörmander operators, for a suitable class of open sets Ω , we find necessary and sufficient conditions on X for the existence of a Lie group (Ω , ∗) such that the operator L = ∑ i = 1 m X i 2 + X 0 is left-invariant with respect to the operation ∗ . Our approach is constructive, as the group law is constructed by means of the solution of a suitable ODE naturally associated to vector fields in X. We provide an application to a partial differential operator appearing in the Finance. [ABSTRACT FROM AUTHOR]

In this paper we develop a comprehensive study on principal eigenvalues and both the (weak and strong) maximum and comparison principles related to an important class of nonlinear systems involving fractional m-Laplacian operators. Explicit lower bounds for principal eigenvalues of this system in terms of the diameter of bounded domain Ω ⊂ R N are also proved. As application, we measure explicitly how small has to be diam (Ω) so that weak and strong maximum principles associated to this problem hold in Ω . [ABSTRACT FROM AUTHOR]

We study the numerical solutions for a class of quasilinear Schrödinger equations arising from the self-channeling of high-power ultra short lasers in matter, which are associated with energy functionals with nonlinear principle parts so that the classical algorithm cannot directly be used. By the method of variable replacement to transform the quasilinear Schrödinger equation into an semilinear elliptic equation, the numerical mountain pass algorithm is then applied. Some numerical experiments are also performed, including zero potential, nonzero constant potential and singular problem. [ABSTRACT FROM AUTHOR]

In this paper, we establish a positive lower bound estimate for the second smallest eigenvalue of the complex Hessian of solutions to a degenerate complex Monge–Ampère equation. As a consequence, we find that in the space of Kähler potentials any two points close to each other in C 2 norm can be connected by a geodesic along which the associated metrics do not degenerate. [ABSTRACT FROM AUTHOR]

With the help of De Giorgi's method, the embedding of the $ \mathfrak {B}_{m(x)}(\Omega,M,\gamma,\gamma _1,\delta,\frac {1}{q}) $ B m (x) (Ω , M , γ , γ 1 , δ , 1 q) class into $ C^{0,\alpha }(\Omega) $ C 0 , α (Ω) is proved. This result can be applied to the $ C^{0,\alpha } $ C 0 , α continuity to some elliptic equations with variable exponent. [ABSTRACT FROM AUTHOR]

For each k = 0 , ⋯ , n we construct a continuous phase f k , with f k (0) = (n - 2 k) π 2 , and viscosity sub- and supersolutions v k , u k , of the elliptic PDE ∑ i = 1 n arctan (λ i (H w)) = f k (x) such that v k - u k has an isolated maximum at the origin. It has been an open question whether the comparison principle would hold in this second order equation for arbitrary continuous phases f : R n ⊇ Ω → (- n π / 2 , n π / 2) . Our examples show it does not. [ABSTRACT FROM AUTHOR]

We establish an Amann-Zehnder-type result for resonance systems of quasilinear elliptic equations with homogeneous Dirichlet boundary conditions, involving nonlinearities growing asymptotically (p,q)\left(p,q)-linear at infinity. The proof relies on a cohomological linking in a product Banach space where the properties of cones of the sublevels are missing, differently from the single quasilinear equation. We also perform critical group computations of the energy functional at the origin, in spite of the lack of its C2{C}^{2} regularity, to exclude that the found mini-max solution is trivial. Finally, we furnish a local condition which guarantees that the found solution is not semi-trivial.

In this study, we investigate some existence results for double phase anisotropic variational problems involving critical growth. We first establish a Lions-type concentration-compactness principle and its variant at infinity for the solution space, which are our independent interests. Using these results, we obtain a nontrivial nonnegative solution to problems of generalized concave-convex type. We also obtain infinitely many solutions when the nonlinear term is symmetric. Our results are new even for the p(⋅)p\left(\cdot )-Laplace equations.

In this paper, we study the unique continuation property for the fractional p-elliptic equations in a semigroup form with variable coefficients. By employing an extension procedure, we derive a monotonicity formula for an extended frequency function. Utilizing this monotonicity together with a blow-up analysis, we establish the unique continuation property. [ABSTRACT FROM AUTHOR]

We consider viscosity solutions to two-phase free boundary problems for the p(x)-Laplacian with non-zero right hand side. We prove that flat free boundaries are C 1 , γ . No assumption on the Lipschitz continuity of solutions is made. These regularity results are the first ones in literature for two-phase free boundary problems for the p(x)-Laplacian and also for two-phase problems for singular/degenerate operators with non-zero right hand side. They are new even when p (x) ≡ p , i.e., for the p-Laplacian. The fact that our results hold for merely viscosity solutions allows a wide applicability. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of infinitely many nontrivial solutions to the following semilinear degenerate elliptic equation - Δ λ u + V (x) u = f (x , u) in R N , N ≥ 3 , where V : R N → R is a potential function and allowed to be vanishing at infinitely, f : R N × R → R is a given function and Δ λ is the strongly degenerate elliptic operator. Under suitable assumptions on the potential V and the nonlinearity f, some results on the multiplicity of solutions are proved. The proofs are based on variational methods, in particular, on the well-known mountain pass lemma of Ambrosetti–Rabinowitz. Due to the vanishing potentials and degeneracy of the operator, some new compact embedding theorems are used in the proof. Our results extend and generalize some existing results (Alves and Souto in J Differ Equ 254:1977–1991, 2013; Hamdani in Asia-Eur J Math 13:2050131, , 2020; Luyen in Commun Math Anal 22:61–75, 2019; Luyen and Tri in J Math Anal Appl 461:1271–1286, 2018; Tang in J Math Anal Appl 401:407–415, 2013; Toon and Ubilla in Discrete Contin Dyn Syst 40:5831–5843, 2020). [ABSTRACT FROM AUTHOR]

In this paper, we investigate sharp singular Trudinger–Moser inequalities involving the anisotropic Dirichlet norm ∫ R N F N (∇ u) d x 1 N in Sobolev-type space D N , q (R N) , N ≥ 2 , q ≥ 1 . Here F : R N → [ 0 , + ∞) is a convex function of class C 2 (R N \ { 0 }) , which is even and positively homogeneous of degree 1. Combing with the connection between convex symmetrization and Schwarz symmetrization, we will establish anisotropic singular Trudinger–Moser inequalities and discuss their sharpness under different situations, including the case ‖ F (∇ u) ‖ N ≤ 1 , the case ‖ F (∇ u) ‖ N a + ‖ u ‖ q b ≤ 1 , and whether they are associated with exact growth. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Bianchi-Egnell inequality on the Heisenberg group, which quantifies the stability of the Sobolev inequality. We generalize results of K o ¨ nig in the Euclidean case. [ABSTRACT FROM AUTHOR]

In this article, we consider a class of degenerate singular problems. The degeneracy is captured by the presence of a class of p-admissible weights, which may vanish or blow up near the origin. Further, the singularity is allowed to vary inside the domain. We provide sufficient conditions on the weight function, on the singular exponent and the source function to establish regularity and existence results. [ABSTRACT FROM AUTHOR]

In a bounded open domain $ \Omega \subset \mathbb {R}^{N} $ Ω ⊂ R N , where $ N\geq 2 $ N ≥ 2 , with Lipschitz boundary $ \partial \Omega $ ∂Ω , we consider the Dirichlet problem for the elliptic system given by \[ \left \{ \begin{array}{@{}ll} \displaystyle-\sum^{N}_{i=1}D_{i}\left(a_{i}(x,u(x),D_{i}u(x))\right)+F(x,u)=f(x), & x\quad \in \Omega,\\ u(x)=0, & x\quad \in \partial \Omega, \end{array} \right. \] { − ∑ i = 1 N D i (a i (x , u (x) , D i u (x))) + F (x , u) = f (x) , x ∈ Ω , u (x) = 0 , x ∈ ∂Ω , here, $ u:\Omega \rightarrow \mathbb {R}^{n} $ u : Ω → R n , $ n\geq 2 $ n ≥ 2 , represents a vector-valued function, $ D_{i}u=\frac {\partial u}{\partial x_{i}} $ D i u = ∂ u ∂ x i denotes the partial derivative of u with respect to $ x_i $ x i , and the vector fields $ a_i:\Omega \times \mathbb {R}^{n}\times \mathbb {R}^{n} \rightarrow \mathbb {R}^{n} $ a i : Ω × R n × R n → R n and $ F:\Omega \times \mathbb {R}^n\rightarrow \mathbb {R}^n $ F : Ω × R n → R n are Carathédory functions. In this paper, we focus on nonlinear degenerate anisotropic elliptic systems with variable growth and $ L^{m} $ L m data, where m is small. Specifically, we consider the case where the right-hand side term f belongs to $ L^{m}(\Omega ;\mathbb {R}^{n}) $ L m (Ω ; R n) with 1

We exploit the existence and non-existence of positive solutions to the eigenvalue problem driven by the nonhomogeneous fractional $ p\& q $ p &q Laplacian operator with indefinite weights \[ \left(-\Delta_p\right)^{\alpha}u + \left(-\Delta_q\right)^{\beta}u = \lambda\left[a \left|u\right|^{p-2}u + b \left|u\right|^{q-2}u \right]\quad{\rm in}\ \Omega, \] (− Δ p) α u + (− Δ q) β u = λ [ a | u | p − 2 u + b | u | q − 2 u ] in Ω , where $ \Omega \subseteq \mathbb {R}^N $ Ω ⊆ R N is a smooth bounded domain that has been extended by zero. We further show the existence of a continuous family of eigenvalues in the case $ \Omega =\mathbb {R}^N $ Ω = R N and $ b\equiv 0 $ b ≡ 0 a.e. Our approach relies strongly on variational Analysis, in which the Mountain pass theorem plays the key role. Due to the lack of spatial compactness and the embedding $ \mathcal {W}^{\alpha, p}\left (\mathbb {R}^N\right) \hookrightarrow \mathcal {W}^{\beta, q}\left (\mathbb {R}^N\right) $ W α , p (R N) ↪ W β , q (R N) in $ \mathbb {R}^N $ R N , we employ the concentration-compactness principle of P.L. Lions [The concentration-compactness principle in the calculus of variations. The limit case. II, Rev Mat Iberoamericana. 1985;1(2):45–121]. to overcome the difficulty. Our paper can be considered as a counterpart to the important works [Alves et al. Existence, multiplicity and concentration for a class of fractional $ p\& q $ p &q Laplacian problems in $ \Bbb R^N $ R N , Commun Pure Appl Anal, 2019;18(4):2009–2045], [Benci et al. An eigenvalue problem for a quasilinear elliptic field equation. J Differ Equ, 2002;184(2):299–320], [Bobkov et al. On positive solutions for $ (p,q) $ (p , q) -Laplace equations with two parameters, Calc Var Partial Differ Equ, 2015;54(3):3277–3301], [Colasuonno and Squassina. Eigenvalues for double phase variational integrals, Ann Mat Pura Appl (4), 2016;195(6):1917–1956], [Papageorgiou et al. Positive solutions for nonlinear Neumann problems with singular terms and convection, J Math Pures Appl (9), 2020;136:1–21], [Papageorgiou et al. Ground state and nodal solutions for a class of double phase problems, Z Angew Math Phys, 2020;71:1–15], and may have further applications to deal with other problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the first eigenvalue λ 1 ⁢ (Ω) of the Grushin operator Δ G := Δ x 1 + | x 1 | 2 ⁢ s ⁢ Δ x 2 with Dirichlet boundary conditions on a bounded domain Ω of ℝ d = ℝ d 1 + d 2 . We prove that λ 1 ⁢ (Ω) admits a unique minimizer in the class of domains with prescribed finite volume, which are the cartesian product of a set in ℝ d 1 and a set in ℝ d 2 , and that the minimizer is the product of two balls Ω 1 * ⊆ ℝ d 1 and Ω 2 * ⊆ ℝ d 2 . Moreover, we provide a lower bound for | Ω 1 * | and for λ 1 ⁢ (Ω 1 * × Ω 2 *) . Finally, we consider the limiting problem as s tends to 0 and to + ∞ . [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear discontinuous sub-elliptic system with drift on the Heisenberg group, where the coefficients in system are discontinuous and satisfy the vanishing mean oscillation condition and growth conditions with the growth index $ 2 2 < p < ∞ , and the non-homogeneous terms satisfy the controllable growth conditions and the natural growth conditions, respectively. The partial Hölder regularity to weak solutions and the partial Morrey regularity to horizontal gradients of weak solutions are proved by the $ \mathcal {A} $ A -harmonic approximation. One of the difficulties in this paper is how to reasonably define the weak solution and avoid the requirement of integrability of Tu. In fact, we used $ T={X_i}{X_{n+i}}-{X_{n+i}}{X_i} $ T = X i X n + i − X n + i X i . [ABSTRACT FROM AUTHOR]

As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type - div ρ a A ∇ w = ρ a f + div ρ a F in Ω for exponents a > - 1 , where the weight ρ vanishes with non zero gradient on a regular hypersurface Γ , which can be either a part of the boundary of Ω or mostly contained in its interior. As an application, we extend such estimates to the ratio v/u of two solutions to a second order elliptic equation in divergence form when the zero set of v includes the zero set of u which is not singular in the domain (in this case ρ = u , a = 2 and w = v / u ). We prove first the C k , α -regularity of the ratio from one side of the regular part of the nodal set of u in the spirit of the higher order boundary Harnack principle in Savin (Discrete Contin Dyn Syst 35–12:6155–6163, 2015). Then, by a gluing Lemma, the estimates extend across the regular part of the nodal set. Finally, using conformal mapping in dimension n = 2 , we provide local gradient estimates for the ratio, which hold also across the singular set. [ABSTRACT FROM AUTHOR]