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In this paper, we introduce quadrature domains for the Helmholtz equation. We show existence results for such domains and implement the so-called partial balayage procedure. We also give an application to inverse scattering problems, and show that there are non-scattering domains for the Helmholtz equation at any positive frequency that have inward cusps. [ABSTRACT FROM AUTHOR]

We study the family of integral equations, called the Robin mean value equations (RMV), that are local averaged approximations to the Robin-Laplace boundary value problem (RL). When posed on C 1 , 1 -regular domains, we prove existence, uniqueness, equiboundedness and the comparison principle for solutions to (RMV). For the continuous right hand side of (RL), we show that solutions to (RMV) converge uniformly, in the limit of the vanishing radius of averaging, to the unique W2,p solution, which coincides with the unique viscosity solution of (RL). We further prove the lower bound on solutions to (RMV), which is consistent with the optimal lower bound for solutions to (RL). Our proofs employ martingale techniques, where (RMV) is interpreted as the dynamic programming principle along a suitable discrete stochastic process, interpolating between the reflecting and the stopped-at-exit Brownian walks. [ABSTRACT FROM AUTHOR]

In this paper, we find some superharmonic functions, which relate to the convexity estimates for Green's function and the first eigenfunction of Laplace operator with homogeneous Dirichlet boundary conditions in bounded convex domains of Rn. [ABSTRACT FROM AUTHOR]

We show that solutions to the Robin mean value equations (RMV), introduced in Lewicka and Peres (2022), converge uniformly in the limit of the vanishing radius of averaging, to the unique solution of the Robin-Laplace boundary value problem (RL), posed on any C 1 , 1 -regular domain and with any bounded Borel right hand side. When compared with the case of continuous right hand side, analyzed in Lewicka and Peres (2022), the present more general setting presents significant technical challenges. Along the way, we prove the asymptotic Hölder equicontinuity of solutions to (RMV): Lipschitz in the interior and C 0 , α up to the boundary, for any α ∈ (0,1). Our proofs employ martingale techniques, where (RMV) is interpreted as the dynamic programming principle for a discrete stochastic process, interpolating between the reflecting and the stopped-at-exit Brownian walks. [ABSTRACT FROM AUTHOR]

We investigate the doubling inequality and upper bounds of nodal sets for Robin and Neumann eigenfunctions on the boundary and in the interior of the domain. Most efforts are devoted to the sharp boundary doubling inequality with new and novel quantitative global Carleman estimates. We are able to obtain the sharp upper bounds for boundary nodal sets of Neumann eigenfunctions. [ABSTRACT FROM AUTHOR]

We consider the torsion function for the Dirichlet Laplacian −Δ, and for the Schrödinger operator −Δ + V on an open set Ω ⊂ ℝ m of finite Lebesgue measure 0 < | Ω | < ∞ with a real-valued, non-negative, measurable potential V. We investigate the efficiency and the phenomenon of localisation for the torsion function, and their interplay with the geometry of the first Dirichlet eigenfunction. [ABSTRACT FROM AUTHOR]

A classical theorem of Bôcher says a positive harmonic function on a punctured domain O ∖ p in Euclidean space can be written as the sum of a constant multiple of the Green function with pole at p and a function harmonic on all of O . Here we establish such a result in the setting of functions on a Riemannian manifold with rather rough metric tensor. [ABSTRACT FROM AUTHOR]

For possibly discontinuous functions including, for instance, Sobolev functions, we present new Blaschke-Privaloff-type criteria for superharmonicity and harmonicity. This opens the way for introduction of a substantial generalization of the Laplace operator. These potential-theoretic considerations lead to a new kind of non-absolutely convergent integral where the integrand may be a highly oscillating pointwise function or even a distribution-valued function. In turn, this integral gives a precise meaning to some generalized Poisson equations with a wild right hand side. [ABSTRACT FROM AUTHOR]

Let D be a non-empty open subset of ℝ m , m ≥ 2 , with boundary ∂D, with finite Lebesgue measure |D|, and which satisfies a parabolic Harnack principle. Let K be a compact, non-polar subset of D. We obtain the leading asymptotic behaviour as ε↓ 0 of the L ∞ norm of the torsion function with a Neumann boundary condition on ∂D, and a Dirichlet boundary condition on ∂(εK), in terms of the first eigenvalue of the Laplacian with corresponding boundary conditions. These estimates quantify those of Burdzy, Chen and Marshall who showed that D ∖ K is a non-trap domain. [ABSTRACT FROM AUTHOR]

Carbery proved that if u : ℝ n → ℝ is a positive, strictly convex function satisfying det D 2 u ≥ 1 , then we have the estimate x ∈ ℝ n : u (x) ≤ s ≲ n s n / 2 and this is optimal. We give a short proof that also implies other results. Our main result is an estimate for the sublevel set of functions u : [ 0 , 1 ] 2 → ℝ satisfying 1 ≤Δu ≤ c for some universal constant c: for any α > 0, we have x ∈ [ 0 , 1 ] 2 : | u (x) | ≤ ε ≲ c ε + ε α − 1 2 ∫ [ 0 , 1 ] 2 | ∇ u | | u | α d x. For 'typical' functions, we expect the integral to be finite for α < 1. While Carbery-Christ-Wright have shown that no sublevel set estimates independently of u exist, this result shows that for 'typical' functions satisfying Δ u ∼ 1 , we expect the sublevel set to be ≲ ε 1 / 2 − . We do not know whether this is sharp or whether similar statements are true in higher dimensions. [ABSTRACT FROM AUTHOR]

In this paper we study the Dirichlet-to-Neumann map for solutions to mean value formulas on trees. We give two alternative definition of the Dirichlet-to-Neumann map. For the first definition (that involves the product of a "gradient" with a "normal vector") and for a linear mean value formula on the directed tree (taking into account only the successors of a given node) we obtain that the Dirichlet-to-Neumann map is given by g ↦ c g ′ (here c is an explicit constant). Notice that this is a local operator of order one. We also consider linear undirected mean value formulas (taking into account not only the successors but the ancestor and the successors of a given node) and prove a similar result. For this kind of mean value formula we include some existence and uniqueness results for the associated Dirichlet problem. Finally, we give an alternative definition of the Dirichlet-to-Neumann map (taking into account differences along a given branch of the tree). With this alternative definition, for a certain range of parameters, we obtain that the Dirichlet-to-Neumann map is given by a nonlocal operator (as happens for the classical Laplacian in the Euclidean space). [ABSTRACT FROM AUTHOR]

The torsion function of a convex planar domain Ω has convex level sets, but explicit formulae are known only for rectangles and ellipses. Here we study the torsion function when the eccentricity of the domain is large. We obtain an approximation for the torsion function on any convex planar domain by viewing the domain as a perturbation of a rectangle in order to define an approximate Green's function for the Laplacian. For a class of convex domains we use this approximation to establish sharp bounds on the Hessian and the infinitesimal shape of the level sets around its maximum. We also use these results to construct examples demonstrating contrasting behaviour of the torsion function and the first eigenfunction of the Dirichlet Laplacian around their respective maxima. [ABSTRACT FROM AUTHOR]

The Khavinson-Shapiro conjecture states that ellipsoids are the only bounded domains in euclidean space satisfying the following property (KS): the solution of the Dirichlet problem for polynomial data is polynomial. In this paper we show that property (KS) for a domain Ω is equivalent to the surjectivity of a Fischer operator associated to the domain Ω. [ABSTRACT FROM AUTHOR]

We give a representation of the solution of the Neumann problem for the Laplace operator on the n-dimensional unit ball in terms of the solution of an associated Dirichlet problem. The representation is extended to other operators besides the Laplacian, to smooth simply connected planar domains, and to the infinite-dimensional Laplacian on the unit ball of an abstract Wiener space, providing in particular an explicit solution for the Neumann problem in this case. As an application, we derive an explicit formula for the Dirichlet-to-Neumann operator, which may be of independent interest. [ABSTRACT FROM AUTHOR]

Let n ≥ 2, let Ω ⊂ ℝ be a bounded domain with $\mathcal {C}^{1}$ boundary, and let $1 \leq p < \frac {2n}{n-2}$ (simply p ≥ 1 if n = 2). The well-known Sobolev imbedding theorem and Rellich compactness implies that is a finite, positive number, and the infimum is achieved by a nontrivial extremal function u, which one can assume is positive inside Ω. We prove that, for 1≤ p ≤ 2 and for every q> p, there exists $K= K(n, p, q, \mathcal {C}_{p}(\Omega ))>0$ such that ∥ u∥ ≥ K∥ u∥. This inequality, which reverses the classical Hölder inequality, mirrors results of G. Chiti for the first Dirichlet eigenfunction of the Laplacian and of M. van den Berg for the torsion function. [ABSTRACT FROM AUTHOR]

The mixed problem is to find a harmonic function having prescribed Dirichlet data on one part of the boundary and prescribed Neumann data on the remainder. One must make a choice as to the required boundary regularity of solutions. When only weak regularity conditions are imposed, the mixed problem has been solved on smooth domains in the plane by Wendland et al. (Math Methods Appl Sci 1(3):265-321, ). Significant advances were later made on Lipschitz domains by Ott and Brown () and Brown (Commun Partial Differ Equ 19(7-8):1217-1233, ). The strain of requiring a square-integrable gradient on the boundary, however, forces a strong geometric restriction on the domain. Well-known counterexamples by Brown show this restriction to be a necessary condition. This paper shows that these counterexamples are an anomaly, in that the mixed problem in the plane can be solved for all data modulo a finite dimensional subspace. The geometric restriction now required is significantly less stringent than the one referred to above. This result is proved by representing solutions in terms of single and double layer potentials, establishing a mixed Rellich inequality, and applying functional analytic arguments to solve a two-by-two system of equations. These results are then extended to allow Robin data in place of Neumann data. [ABSTRACT FROM AUTHOR]

We prove continuity properties for the Oseen potential. As a consequence, we show some new properties on solutions of the Oseen equations. The study relies on weighted Sobolev spaces in order to control the behavior of functions at infinity. [ABSTRACT FROM AUTHOR]

In this paper we consider the stochastic Dirichlet problem $L\lozenge u=h+\nabla f$ in the framework of white noise analysis combined with Sobolev space and Colombeau algebra methods. The operator L is assumed to be strictly elliptic in divergence form $L\lozenge u=\nabla(A\lozenge\nabla u+b\lozenge u)+c\lozenge\nabla u+d\lozenge u$. Its coefficients: the elements of the matrix A and of the vectors b, c and d are assumed to be generalized random processes, and the product of two generalized processes is interpreted as the Wick product. Generalized random processes are considered as linear bounded mappings from the Sobolev space $W_0^{1,2}$ into the Kondratiev space ( S). In this paper we prove existence and uniqueness of the problem of this form in the case when the operator L generates a coercive bilinear form, and then extend this result to the general case. We also consider the case when the coefficients of L, the input data and the boundary condition are Colombeau-type generalized stochastic processes. [ABSTRACT FROM AUTHOR]

We treat the stochastic Dirichlet problem $L\lozenge u = h+\nabla f$ in the framework of white noise analysis combined with Sobolev space methods. The input data and the boundary condition are generalized stochastic processes regarded as linear continuous mappings from the Sobolev space $W_0^{1,2}$ into the Kondratiev space ( S) − 1. The operator L is assumed to be strictly elliptic in divergence form $L\lozenge u=\nabla(A\lozenge\nabla u+b\lozenge u)+c\lozenge\nabla u+d\lozenge u$. Its coefficients: the elements of the matrix A and of the vectors b, c and d are assumed to be generalized random processes, and the product of two generalized processes, denoted by $\lozenge$, is interpreted as the Wick product. In this paper we prove the weak maximum principle for the operator L, which will imply the uniqueness of the solution to $L\lozenge u = h+\nabla f$. [ABSTRACT FROM AUTHOR]

Let μ be a positive measure with finite support in ℜ ( n≧2). Then we show that there is a bounded open set Ω, containing the support of μ, whose single-layer potential coincides with the potential of μ outside Ω. [ABSTRACT FROM AUTHOR]

We study the first eigenfunction φ of the Dirichlet Laplacian on a convex domain in Euclidean space. Elementary properties of Bessel functions yield that $$\left\| {\phi _1 } \right\|_\infty /\left\| {\phi _1 } \right\|_2 \to \infty$$ if D is a sector in Euclidean plane with area 1 and the angle tends to 0. We aim to characterize those domains D such that $$(vol(D))^{1/2} \left\| {\phi _1 } \right\|_\infty /\left\| {\phi _1 } \right\|_2$$ is large in terms of the ratio of the first eigenvalue of D and the infimum of the first eigenvalues of all subdomains D≈ of D with given volume. [ABSTRACT FROM AUTHOR]