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In this paper, we study those mappings in unit ball satisfying the Dirichlet problem of the following differential operators Δ γ = (1 - | x | 2) · 1 - | x | 2 4 · ∑ i ∂ 2 ∂ x i 2 + γ ∑ i x i · ∂ ∂ x i + γ n 2 - 1 - γ. Our aim is to establish the Schwarz type inequality, Heinz-Schwarz type inequality and boundary Schwarz inequality for those mappings. [ABSTRACT FROM AUTHOR]

In this paper, when studying the connection between the fractional convexity and the fractional p-Laplace operator, we deduce a nonlocal and nonlinear equation. Firstly, we will prove the existence and uniqueness of the viscosity solution of this equation. Then we will show that u ⁢ (x) is the viscosity sub-solution of the equation if and only if u ⁢ (x) is so-called (α , p) -convex. Finally, we will characterize the viscosity solution of this equation as the envelope of an (α , p) -convex sub-solution. The technique involves attainability of the exterior datum and a comparison principle for the nonlocal and nonlinear equation. [ABSTRACT FROM AUTHOR]

Let uu be a nontrivial harmonic function in a domain D⊂RdD\subset {{\mathbb{R}}}^{d}, which vanishes on an open set of the boundary. In a recent article, we showed that if DD is a C1{C}^{1}-Dini domain, then, within the open set, the singular set of uu, defined as {X∈D¯:u(X)=0=∣∇u(X)∣}\left\{X\in \overline{D}:u\left(X)=0=| \nabla u\left(X)| \right\}, has finite (d−2)\left(d-2)-dimensional Hausdorff measure. In this article, we show that the assumption of C1{C}^{1}-Dini domains is sharp, by constructing a large class of non-Dini (but almost Dini) domains whose singular sets have infinite ℋd−2{{\mathcal{ {\mathcal H} }}}^{d-2}-measures.

The aim of this article is twofold. The first goal is to give a new characterization of the Kato class of functions K ∞ (R + d) that was defined in (Bachar et al. 2002:41, 2002) for d = 2 and in (Bachar and Mâagli 9(2):153–192, 2005) for d ≥ 3 and adapted to study some nonlinear elliptic problems in the half space. The second goal is to prove the existence of positive continuous weak solutions, having the global behavior of the associated homogeneous problem, for sufficiently small values of the nonnegative constants λ and μ to the following system: Δ u = λ f (x , u , v) , Δ v = μ g (x , u , v) in R + d , lim x → (ξ , 0) u (x) = a 1 ϕ 1 (ξ) , lim x → (ξ , 0) v (x) = a 2 ϕ 2 (ξ) for all ξ ∈ R d − 1 , lim x d → ∞ u (x) x d = b 1 , lim x d → ∞ v (x) x d = b 2 , where ϕ 1 and ϕ 2 are nontrivial nonnegative continuous functions on ∂ R + d = R d − 1 × { 0 } , a 1 , a 2 , b 1 , b 2 are nonnegative constants such that (a 1 + b 1) (a 2 + b 2) > 0 . The functions f and g are nonnegative and belong to a class of functions containing in particular all functions of the type f (x , u , v) = p (x) u α g 1 (v) and g (x , u , v) = q (x) g 2 (u) v β with α ≥ 1 , β ≥ 1 , g 1 , g 2 are continuous on [ 0 , ∞) , and p , q are nonnegative functions in K ∞ (R + d) . [ABSTRACT FROM AUTHOR]

Questions concerning quantitative and asymptotic properties of the elliptic measure corresponding to a uniformly elliptic divergence form operator have been the focus of recent studies. In this setting we show that the elliptic measure of an operator with coefficients satisfying a vanishing Carleson condition in the upper half space is an asymptotically optimal A ∞ weight. In particular, for such operators the logarithm of the elliptic kernel is in the space of (locally) vanishing mean oscillation. To achieve this, we prove local, quantitative estimates on a quantity (introduced by Fefferman, Kenig and Pipher) that controls the A ∞ constant. Our work uses recent results obtained by David, Li and Mayboroda. These quantitative estimates may offer a new framework to approach similar problems. [ABSTRACT FROM AUTHOR]

Let Ω be a bounded domain in ℝ n . Under appropriate conditions on Ω, we prove existence and uniqueness of continuous functions solving the Dirichlet problem associated to certain nonlinear mean value properties in Ω with respect to balls of variable radius. We also show that, when properly normalized, such functions converge to the p-harmonic solution of the Dirichlet problem in Ω for p ⩾ 2 . Existence is obtained via iteration, a fundamental tool being the construction of explicit universal barriers in Ω. [ABSTRACT FROM AUTHOR]

This article provides a functional analytical framework for boundary integral equations of the heat equation in time-dependent domains. More specifically, we consider a non-cylindrical domain in space-time that is the C 2 -diffeomorphic image of a cylinder, i.e., the tensor product of a time interval and a fixed domain in space. On the non-cylindrical domain, we introduce Sobolev spaces, trace lemmata and provide the mapping properties of the layer operators. Here it is critical that the Neumann trace requires a correction term for the normal velocity of the moving boundary. Therefore, one has to analyze the situation carefully. [ABSTRACT FROM AUTHOR]

Let Ω ⊆ R n be an open set, where n ≥ 2 . Suppose ω is a locally finite Borel measure on Ω . For α ∈ (0 , 2) , define the fractional Laplacian (- ▵) α / 2 via the Fourier transform on R n , and let G be the corresponding Green's operator of order α on Ω . Define T (u) = G (u ω). If ‖ T ‖ L 2 (ω) → L 2 (ω) < 1 , we obtain a representation for the unique weak solution u in the homogeneous Sobolev space L 0 α / 2 , 2 (Ω) of (- ▵) α / 2 u = u ω + ν on Ω , u = 0 on Ω c , for ν in the dual Sobolev space L - α / 2 , 2 (Ω) . If Ω is a bounded C 1 , 1 domain, this representation yields matching exponential upper and lower pointwise estimates for the solution when ν = χ Ω . These estimates are used to study the existence of a solution u 1 (called the "gauge") of the integral equation u 1 = 1 + G (u 1 ω) corresponding to the problem (- ▵) α / 2 u = u ω on Ω , u ≥ 0 on Ω , u = 1 on Ω c. We show that if ‖ T ‖ < 1 , then u 1 always exists if 0 < α < 1 . For 1 ≤ α < 2 , a solution exists if the norm of T is sufficiently small. We also show that the condition ‖ T ‖ < 1 does not imply the existence of a solution if 1 < α < 2 . The condition ‖ T ‖ ≤ 1 is necessary for the existence of u 1 for all 0 < α ≤ 2 . [ABSTRACT FROM AUTHOR]

In this paper, a Cauchy problem of Helmholtz equation in a multi-dimensional case is investigated. This problem is severely ill-posed and small perturbations to measurement data can result in large changes in the solution. A kind of operator regularization method is proposed. The stable error estimates are obtained in the L 2 − norm and H r − norm under the conditions that m is even, md>x, mk>1, and suitable choice of regular parameters. Error estimates show that the regularized solution depends continuously on the perturbation noisy data and wave number. Our method makes up the limitation of small waves. Three numerical experiments show that our proposed method is effective and stable, especially in the case of a large wave number. [ABSTRACT FROM AUTHOR]

The present paper establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition in uniform domains with Ahlfors regular boundaries. The result can be viewed as a quantitative analogue of the Wiener criterion adapted to the singular L p data case. The first step is taken in Part I, where we considered the case in which the desired Carleson measure condition on the coefficients holds with sufficiently small constant, using a novel application of techniques developed in geometric measure theory. In Part II we establish the final result, that is, the "large constant case". The key elements are a powerful extrapolation argument, which provides a general pathway to self-improve scale-invariant small constant estimates, and a new mechanism to transfer quantitative absolute continuity of elliptic measure between a domain and its subdomains. [ABSTRACT FROM AUTHOR]

In this work, we study Schrödinger type equations with domain in a complex space and their solutions in the classical as well as in the sense of the space of currents. For such an equation with a (possibly) complex parameter and a (possibly) nonzero potential, conditions are given under which an analogue of the Weyl's lemma holds, meaning that each solution within the space of currents is distributionally realizable (in a suitable sense) by a semismooth, or by a semi-real-analytic function. Motivated by the Schwartz–Gunning theorem on the holomorphy of a locally integrable function, conditions guaranteeing a regular current on a general domain to be realized by a function which is (locally) a solution of the Schrödinger equation are obtained. Also, generalized Helmholtz formulas are derived and utilized in studying some special classes of zero-degree currents in regard to the property of μ-harmonicity or weak-harmonicity. [ABSTRACT FROM AUTHOR]

We consider second-order divergence form elliptic operators with W1,1 coefficients, in a uniform domain Ω with Ahlfors regular boundary. We show that the A∞ property of the elliptic measure associated to any such operator implies that Ω is a set of locally finite perimeter whose boundary, ∂ ⁡Ω, is rectifiable. As a corollary we show that for this type of operators, absolute continuity of the surface measure with respect to the elliptic measure is enough to guarantee rectifiability of the boundary. In the case that the coefficients are continuous we obtain additional information about Ω. [ABSTRACT FROM AUTHOR]

In this paper, two Cauchy problems of Helmholtz equation in a three-dimensional case are considered. To address these problems, a mollification method with bivariate Dirichlet kernel is proposed. Stable errors estimates are obtained based on appropriate a priori choices of mollification parameters. Convergence estimates show that the regularization solution depends continuously on the data and wavenumber. Numerical examples of our interest show that Dirichlet kernel is more effective than the Gaussian kernel under the same parameter selection rule, and our procedure is stable with respect to perturbations noise in the data. [ABSTRACT FROM AUTHOR]

We consider the Dirichlet boundary value problem for divergence form elliptic operators with bounded measurable coefficients. We prove that for uniform domains with Ahlfors regular boundary, the BMO solvability of such problems is equivalent to a quantitative absolute continuity of the elliptic measure with respect to the surface measure, i.e., ωL∈A∞(σ). This generalizes a previous result on Lipschitz domains by Dindos, Kenig, and Pipher (see Dindos et al. in J Geom Anal 21:78-95, 2011). [ABSTRACT FROM AUTHOR]

We establish a general concept of families of hypersurfaces over which the integral of a harmonic function remains invariant. Passing to the limit these hypersurfaces often degenerate, a fact that renders the majority of the classical mean value properties, and also helps to find new ones. [ABSTRACT FROM AUTHOR]

We study the Dirichlet problem for non-homogeneous equations involving the fractional p-Laplacian. We apply Perron's method and prove Wiener's resolutivity theorem. [ABSTRACT FROM AUTHOR]

As a complement of (Shi and Xiao in Potential Anal 45:261-298, 2016) this article addresses five more aspects of the fractional capacities relative to bounded open Lipschitz subsets of $${\mathbb {R}}^n$$ : (i) quasi-everywhere version; (ii) unique extremal; (iii) energy functional; (iv) dual representation; (v) fine behaviour of each fractional Sobolev function in $${\mathbb {R}}^n$$ . [ABSTRACT FROM AUTHOR]

One of the most cited papers in Applied Mechanics is the work of Eshelby from 1957 who showed that a homogeneous isotropic ellipsoidal inhomogeneity embedded in an unbounded (in all directions) homogeneous isotropic host would feel uniform strains and stresses when uniform strains or tractions are applied in the far-field. Of specific importance is the uniformity of Eshelby's tensor $\mathbf{S}$ . Following Eshelby's seminal work, a vast literature has been generated using and developing Eshelby's result and ideas, leading to some beautiful mathematics and extremely useful results in a wide range of application areas. In 1961 Eshelby conjectured that for anisotropic materials only ellipsoidal inhomogeneities would lead to such uniform interior fields. Although much progress has been made since then, the quest to prove this conjecture is still not complete; numerous important problems remain open. Following a different approach to that considered by Eshelby, a closely related tensor $\mathbf{P}=\mathbf{S}\mathbf{D}^{0}$ arises, where $\mathbf{D}^{0}$ is the host medium compliance tensor. The tensor $\mathbf{P}$ is associated with Hill and is of course also uniform when ellipsoidal inhomogeneities are embedded in a homogeneous host phase. Two of the most fundamental and useful areas of applications of these tensors are in Newtonian potential problems such as heat conduction, electrostatics, etc. and in the vector problems of elastostatics. Knowledge of the Hill and Eshelby tensors permit a number of interesting aspects to be studied associated with inhomogeneity problems and more generally for inhomogeneous media. Micromechanical methods established mainly over the last half-century have enabled bounds on and predictions of the effective properties of composite media. In many cases such predictions can be explicitly written down in terms of the Hill tensor, or equivalently the Eshelby tensor and can be shown to provide excellent predictions in many cases. Of specific interest is that a number of important limits of the ellipsoidal inhomogeneity can be taken in order to be employed in predictions of the effective properties of, for example, layered media and fibre reinforced composites and also to the cases when voids and cracks are present. In the main, results for the Hill and Eshelby tensors are distributed over a wide range of articles and books, using different notation and terminology and so it is often difficult to extract the necessary information for the tensor that one requires. The case of an anisotropic host phase is also frequently non-trivial due to the requirement of the associated Green's tensor. Here this classical problem is revisited and a large number of results for problems that are felt to be of great utility in a wide range of disciplines are derived or recalled. A scaling argument leads to the derivation of the Eshelby tensor for potential problems where the host phase is at most orthotropic, without the requirement of using the anisotropic Green's function. The Concentration tensor $\boldsymbol{\mathcal{A}}$ linking interior fields to those imposed in the far-field is derived for a wide variety of problems. These results can therefore be used in the various micromechanical schemes. Directly related to the tensors of Eshelby and Hill is the so-called Moment tensor $\mathbf{M}$ . As well as arising in the literature on micromechanics, this tensor is important in the vast area of research associated with inverse problems and specifically with the problem of identifying an object inside some domain given the application of a specific set of boundary conditions. Due to its fundamental importance and direct link to the Eshelby and Hill tensors, here we state the connection between $\mathbf{M}, \mathbf{P}$ and $\mathbf{S}$ in an effort to ensure that the work is of use to as wide a community as possible. Both tensor and matrix formulations are considered and contrasted throughout. Appendices give various details that illustrate the implementation of both approaches. [ABSTRACT FROM AUTHOR]

For 1≤ p≤ ∞ and n/ p< α<1 + n/ p, the fractional capacity C ( K,Ω) of a compact subset K with respect to a bounded open Lipschitz set Ω⊃ K of $\mathbb R^{n}$ is systematically investigated and effectively applied to the fractional Sobolev space W (Ω) which plays an important role in calculus of variations, harmonic analysis, potential theory, partial differential equations, mathematical physics and so forth. In particular, we explore: [ABSTRACT FROM AUTHOR]

In this paper, we investigate single and double layer potentials mapping boundary data to interior functions of a domain at high frequency λ 2 → ∞ . In the high frequency regime the key problem is the dependence of mapping norms on the parameter λ . For single layer potentials, we find that the L 2 ( ∂ Ω ) → L 2 ( Ω ) norms decay in λ . The rate of decay depends on the curvature of ∂Ω: The norm is λ − 3 / 4 in piecewise smooth domains and λ − 5 / 6 if the boundary ∂Ω is positively curved. The double layer potential, however, displays uniform L 2 ( ∂ Ω ) → L 2 ( Ω ) bounds independent of curvature. By various examples, we show that all our estimates on layer potentials are sharp. Appendix A by Galkowski gives bounds L 2 ( ∂ Ω ) → L 2 ( ∂ Ω ) for the single and double layer operators at high frequency that are sharp modulo log ⁡ λ . In this case, both the single and double layer operator bounds depend upon the curvature of the boundary. [ABSTRACT FROM AUTHOR]

We study higher dimensional counterparts to the well-known theorem of Pavlović (Ann. Acad. Sci. Fenn. Math. 27, 365-372, ), that every harmonic quasiconformal mapping of the disk is bi-Lipschitz. [ABSTRACT FROM AUTHOR]