Narrow Results

In this paper, we study the weak strong uniqueness of the Dirichlet type problems of fractional Laplace (Poisson) equations. We construct the Green’s function and the Poisson kernel. We then provide a somewhat sharp condition for the solution to be unique. We also show that the solution under such condition exists and must be given by our Green’s function and Poisson kernel. In doing these, we establish several basic and useful properties of the Green’s function and Poisson kernel. Based on these, we obtain some further a priori estimates of the solutions. Surprisingly those estimates are quite different from the ones for the local type elliptic equations such as Laplace equations. These are basic properties to the fractional Laplace equations and can be useful in the study of related problems.

In a microscopic scale or microgravity environment, interfaces in wetting phenomena are usually modeled by surfaces with constant mean curvature (CMC surfaces). Usually, the condition regarding the constancy of the contact angle along the line of separation between different phases is assumed. Although the classical capillary boundary condition is the angle made at the contact line, configurations also occur in which a Dirichlet condition is appropriate. In this article, we discuss those with vanishing boundary conditions, such as those that occur on a thin flat portion of a plate of general shape covered with water. In this paper, we review recent works on the existence of CMC surfaces with non-empty boundary, with a special focus on the Dirichlet problem for the constant mean curvature equation.

We give a sharp estimate of the modulus of continuity of the solution to the Dirichlet problem for the complex Hessian equation of order $m$, $1\le m\le n$, with a continuous right-hand side and a continuous boundary data in a bounded strongly $m$-pseudoconvex domain $\mathrm{\Omega }\subset {ℂ}^n$. Moreover, when the right-hand side is in $L^p(\mathrm{\Omega })$ for some $p>n/m$, and the boundary value function is $c^{1,1}$-smooth we prove that the solution is Hölder continuous.

In this paper, we establish existence of Hölder continuous solutions to the complex Monge–Ampère-type equation with measures vanishing on pluripolar subsets of a bounded strictly pseudoconvex domain $Ø$ in ${ℂ}^n$.

This paper is devoted to the study of properties of second-order elliptic equation solutions. The main content of the paper coincides with the report made by the author at the international conference dedicated to the 75th anniversary of I. V. Volovich. The solution behavior near the boundary and the Dirichlet problem formulation, which is closely related to this issue, are studied.

At the end of the paper, we will briefly discuss the results obtained in elegant and extremely important works by E. De Giorgi and J. Nash regarding Hölder continuity of the equation solutions within the considered domain. We present results that combine and complement the belonging of the solution to the Hölder and Sobolev spaces. Note that all the concepts and statements under consideration are united by a common approach and are formulated in close terms.

Considering the Dirichlet problem for Poisson's equation in two and three dimensions, we derive a posteriori error estimators for finite element solutions with interpolated boundary values. The estimators are reliable and (locally) efficient with respect to the energy norm error, also in the case of discontinuous boundary values and load terms that are not square-integrable due to singularities at the boundary of the underlying domain. Moreover, we propose an adaptive algorithm based upon these estimators and test it also in nonsmooth cases of the aforementioned type: its convergence rate is optimal.

This paper studied the level-3 Sierpinski gasket. We solved Dirichlet problem of Poisson equations and proved variational principle on the level-3 Sierpinski gasket by expressing Green’s function explicitly.

There are three over-determined boundary value problems for the inhomogeneous Cauchy–Riemann equation, the Dirichlet problem, the Neumann problem and the Robin problem. These problems have found their significant importance and applications in diverse fields of science, for example, applied mathematics, physics, engineering and medicine. In this paper, we explicitly investigate the solvability of these three basic boundary value problems for the Cauchy–Riemann operator on an isosceles orthogonal triangle. The solvability conditions are explicitly obtained. The plane parqueting reflection principle is used. Cauchy–Pompeiu-type formulae for the triangle are established. The boundary behavior of a related integral operator of the Schwarz type is studied in detail. Then, the boundary values of solutions at the corner points are found.

We give a self-contained treatment of the existence of a regular solution to the Dirichlet problem for harmonic maps into a geodesic ball on which the squared distance function from the origin is strictly convex. No curvature assumptions on the target are required. In this route we introduce a new deformation result which permits to glue a suitable Euclidean end to the geodesic ball without violating the convexity property of the distance function from the fixed origin. We also take the occasion to analyze the relationships between different notions of Sobolev maps when the target manifold is covered by a single normal coordinate chart. In particular, we provide full details on the equivalence between the notions of traced Sobolev classes of bounded maps defined intrinsically and in terms of Euclidean isometric embeddings.

We solve the Dirichlet problem for complex Monge–Ampère equation near an isolate Klt singularity, which generalizes the result of Eyssidieux et al. [Singular Kähler–Einstein metrics, J. Amer. Math. Soc. 22 (2009) 607–639], where the Monge–Ampère equation is solved on singular varieties without boundary. As a corollary, we construct solutions to Monge–Ampère equation with isolated singularity on strongly pseudoconvex domain $\mathrm{\Omega }$ contained in ${ℂ}^n$.

In this article, we consider finite volume methods based on a non-uniform grid. Finite volume methods are compared to finite difference methods based on a related grid. As an application, various convergence results are proved for the finite volume function spaces and for some model elliptic and parabolic boundary value problems using these discretization spaces.

In this paper, by applying the well-known method for dealing with $p$-Laplace type elliptic boundary value problems, the authors establish a sharp estimate for the decreasing rearrangement of the gradient of solutions to the Dirichlet and the Neumann boundary value problems of a class of Schrödinger equations, under the weak regularity assumption on the boundary of domains. As applications, the gradient estimates of these solutions in Lebesgue spaces and Lorentz spaces are obtained.

The objective of this paper is to consider the theory of regularity of systems of partial differential equations with Neumann boundary conditions. It complements previous works of the authors for the Dirichlet case. This type of problem is motivated by stochastic differential games. The Neumann case corresponds to stochastic differential equations with reflection on boundary of the domain.

In this paper, the author studies the regularity of solutions to the Dirichlet problem for equation Lu = f, where L is a second order degenerate elliptic operator in divergence form in Ω, a bounded open subset of Rⁿ(n ≥ 3).

The objective of this paper is to consider the theory of regularity of systems of partial differential equations with Neumann boundary conditions. It complements previous works of the authors for the Dirichlet case. This type of problem is motivated by stochastic differential games. The Neumann case corresponds to stochastic differential equations with reflection on boundary of the domain.

A certain Dirichlet boundary value problem for the polyharmonic operator of arbitrary order n is explicitly solved in case of the unit disc of the complex plane. Bassisc tools are a related Cauchy-Pompeiu representation formula modified with a proper polyharmonic Green function. A similar but asymmetric Dirichlet problem is recently treated with the same method.

A function hconverges controlled by a non negative function k, to a function f if h has finite limits equal to f along those sets where k is bounded and if h/k converges to 0 where k converges to +∞.

The controlled convergence yields a new method for setting and solving the Dirichlet problem for general open sets and general boundary data. It is also a simple and useful tool for studying extensions of linear positive operators defined on spaces of continuous functions, e.g. the solution of the Dirichlet problem or the mean–value operator. Continuity and convergence criteria may be obtained out of boundedness criteria.

The harmonic Dirichlet problem in a planar domain with smooth cracks of an arbitrary shape is considered in case, when the solution is not continuous at the ends the cracks. The well–posed formulation of the problem is given, theorems on existence and uniqueness of a classical solution are proved, the integral representation for a solution is obtained. With the help of the integral representation, the properties of the solution are studied. It is proved that a weak solution of the Dirichlet problem in question does not exist typically, though the classical solution exists.

The Dirichlet problem is studied for elliptic systems in devergence form and discontinuous coefficents

We study the Dirichlet problem for a class of nonlinear elliptic equations with variable exponents of nonlinearity. We prove existence of solutions in a generalized Sobolev–Orlicz space and study the localization (vanishing) properties of the solutions. It is shown that unlike the equations with isotropic nonlinearity the anisotropy may cause localization of solutions in a separate direction. We use this property to establish sufficient conditions of solvability of the problems posed on unbounded domains without conditions at infinity. These condition relate the "asymptotic size" of the domain at infinity with the character of nonlinearity of the equation.