Overdetermined Boundary Value Problems Research Articles

Stability of the Wulff shape with respect to anisotropic curvature functionals

For a function f which foliates a one-sided neighborhood of a closed hypersurface M, we give an estimate of the distance of M to a Wulff shape in terms of the Lp-norm of the traceless F-Hessian of f, where F is the support function of the Wulff shape. This theorem is applied to prove quantitative stability results for the anisotropic Heintze-Karcher inequality, the anisotropic Alexandrov problem, as well as for the anisotropic overdetermined boundary value problem of Serrin-type.

The overdetermined Cauchy problem for the hyperbolic Gellerstedt equation

Abstract Overdetermined boundary value problems and the minimal operators generated by them are extremely important in the description of regular boundary value problems for differential equations, and are also widely used in the study of local properties of solutions. In addition, for inverse problems of mathematical physics arising from applications, when determining unknown data, it is necessary to study problems with overdetermined boundary conditions, which is reflected in the study of problems, including for hyperbolic equations and systems, arising in physics, geophysics, seismic tomography, geoelectrics, electrodynamics, medicine, ecology, economics and many other practical areas. Thus, the study of overdetermined boundary value problems is of both theoretical and applied interest. In this paper, a criterion for the regular solvability of the overdetermined Cauchy problem for the Gellerstedt equation and the minimal differential operator generated by it in a hyperbolic domain is established, as which both the case of a characteristic triangle and the case of a more general domain with fairly general assumptions about the boundary of the domain are considered. Due to overdetermined boundary conditions, the problem under consideration will be ill-posed in the general case, therefore, for its regular solvability, additional conditions must be imposed on the initial data. In other words, we have considered the inverse problem: to determine what requirements the initial data of the problem, in particular the right part of the Gellerstedt equation, should meet, in question, so that the overdetermined Cauchy problem is regularly solvable. The proof is based on the Gellerstedt potential, the properties of solutions of the Goursat problem in the characteristic triangle, and the properties of special functions.

A General Integral Identity with Applications to a Reverse Serrin Problem

We prove a new general differential identity and an associated integral identity, which entails a pair of solutions of the Poisson equation with constant source term. This generalizes a formula that the first and third authors previously proved and used to obtain quantitative estimates of spherical symmetry for the Serrin overdetermined boundary value problem. As an application, we prove a quantitative symmetry result for the reverse Serrin problem, which we introduce for the first time in this paper. In passing, we obtain a rigidity result for solutions of the aforementioned Poisson equation subject to a constant Neumann condition.

Initiation of decohesion between a flat punch and a thin bonded incompressible layer

Non-axisymmetric frictionless JKR-type adhesive contact between a rigid body and a thin incompressible elastic layer bonded to a rigid base is considered in the framework of the leading-order asymptotic model, which has the form of an overdetermined boundary value problem. Based on the first-order perturbation of the Neumann operator in the Dirichlet problem for Poisson’s equation, the decohesion initiation problem is formulated in the form of a variational inequality. The asymptotic model assumes that the contact zone and its boundary contour during the detachment process are unknown. The absence of the solvability theorem is illustrated by an example of the instability of an axisymmetric flat circular contact.

Epsilon-regularity for the solutions of a free boundary system

This paper is dedicated to a free boundary system arising in the study of a class of shape optimization problems. The problem involves three variables: two functions u and v , and a domain \Omega ; with u and v being both positive in \Omega , vanishing simultaneously on \partial\Omega , and satisfying an overdetermined boundary value problem involving the product of their normal derivatives on \partial\Omega . Precisely, we consider solutions u, v \in C(B_1) of -\Delta u= f \ \text{ and } \ -\Delta v=g \quad\text{in }\Omega=\{u>0\}=\{v>0\}, \frac{\partial u}{\partial n}\frac{\partial v}{\partial n}=Q \quad\text{on }\partial\Omega\cap B_1. Our main result is an epsilon-regularity theorem for viscosity solutions of this free boundary system. We prove a partial Harnack inequality near flat points for the couple of auxiliary functions \sqrt{uv} and \frac12(u+v) . Then, we use the gained space near the free boundary to transfer the improved flatness to the original solutions. Finally, using the partial Harnack inequality, we obtain an improvement-of-flatness result, which allows to conclude that flatness implies C^{1,\alpha} regularity.

Overdetermined problems in annular domains with a spherical-boundary component in space forms

We obtain a Serrin-type symmetry of the solutions to various overdetermined boundary value problems in annular domains with a spherical-boundary component in space forms by using the maximum principle for suitable subharmonic functions and integral identities.

On Over-Determined Boundary Value Problems for Hyperbolic Equations

On Over-Determined Boundary Value Problems for Hyperbolic Equations

Radial symmetry and partially overdetermined problems in a convex cone

AbstractWe obtain the radial symmetry of the solution to a partially overdetermined boundary value problem in a convex cone in space forms by using the maximum principle for a suitable subharmonic function P and integral identities. In dimension 2, we prove Serrin‐type results for partially overdetermined problems outside a convex cone. Furthermore, we obtain a Rellich identity for an eigenvalue problem with mixed boundary conditions in a cone.

COMPLEX BOUNDARY VALUE PROBLEMS FOR THE CAUCHY–RIEMANN OPERATOR ON A TRIANGLE

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.

A Carleman-based numerical method for quasilinear elliptic equations with over-determined boundary data and applications

We propose a new iterative scheme to compute the numerical solution to an over-determined boundary value problem for a general quasilinear elliptic PDE. The main idea is to repeatedly solve its linearization by using the quasi-reversibility method with a suitable Carleman weight function. The presence of the Carleman weight function allows us to employ a Carleman estimate to prove the convergence of the sequence generated by the iterative scheme above to the desired solution. The convergence of the iteration is fast at an exponential rate without the need of an initial good guess. We apply this method to compute solutions to some general quasilinear elliptic equations and a large class of first-order Hamilton-Jacobi equations. Numerical results are presented.

Foliation of an Asymptotically Flat End by Critical Capacitors

We construct a foliation of an asymptotically flat end of a Riemannian manifold by hypersurfaces which are critical points of a natural functional arising in potential theory. These hypersurfaces are perturbations of large coordinate spheres, and they admit solutions of a certain over-determined boundary value problem involving the Laplace–Beltrami operator. In a key step we must invert the Dirichlet-to-Neumann operator, highlighting the nonlocal nature of our problem.

Interpolating estimates with applications to some quantitative symmetry results

<abstract><p>We prove interpolating estimates providing a bound for the oscillation of a function in terms of two $ L^p $ norms of its gradient. They are based on a pointwise bound of a function on cones in terms of the Riesz potential of its gradient. The estimates hold for a general class of domains, including, e.g., Lipschitz domains. All the constants involved can be explicitly computed. As an application, we show how to use these estimates to obtain stability for Alexandrov's Soap Bubble Theorem and Serrin's overdetermined boundary value problem. The new approach results in several novelties and benefits for these problems.</p></abstract>

Existence and stability of solutions for a fourth order overdetermined problem

We examine a Serrin-type overdetermined boundary value problem for the biharmonic operator. If the underlying set is the unit ball, a solution exists for a constant overdetermining condition. We prove the existence of an open and bounded domain admitting a solution to the boundary value problem for every small perturbation of the overdetermining condition. Moreover, we establish stability estimates for the deviation of this domain from the unit ball in terms of the perturbation.Our approach is motivated by a recent result of Gilsbach and Onodera and applies a result of Ferrero, Gazzola and Weth for a fourth order Steklov problem.

On fourth order accuracy stable difference scheme for a multi-point overdetermined elliptic problem

In this paper fourth order of accuracy difference scheme for approximate solution of a multi-point elliptic overdetermined problem in a Hilbert space is proposed. The existence and uniqueness of the solution of the difference scheme are obtained by using the functional operator approach. Stability, almost coercive stability, and coercive stability estimates for the solution of difference scheme are established. These theoretical results can be applied to construct a stable highly accurate difference scheme for approximate solution of multi-point overdetermined boundary value problem for multidimensional elliptic partial differential equations.

Symmetry breaking bifurcations for two overdetermined boundary value problems with non-constant Neumann condition on exterior domains in

We study two overdetermined elliptic boundary value problems on exterior domains (the complement of a ball and the complement of a solid cylinder in respectively). The Neumann condition is non-constant and involves the mean curvature of the boundary. We show there exists a family of bifurcation branches of domains which are small deformations of the complement of a ball and of the complement of a solid cylinder, respectively, and which support the solution of the overdetermined boundary value problem.

Remarks on radial symmetry and monotonicity for solutions of semilinear higher order elliptic equations

<abstract><p>Half a century after the appearance of the celebrated paper by Serrin about overdetermined boundary value problems in potential theory and related symmetry properties, we reconsider semilinear polyharmonic equations under Dirichlet boundary conditions in the unit ball of $ \mathbb{R}^{n} $. We discuss radial properties (symmetry and monotonicity) of positive solutions of such equations and we show that, in <italic>conformal dimensions</italic>, the associated Green function satisfies elegant reflection and symmetry properties related to a suitable Kelvin transform (inversion about a sphere). This yields an alternative formula for computing the partial derivatives of solutions of polyharmonic problems. Moreover, it gives some hints on how to modify a counterexample by Sweers where radial monotonicity fails: we numerically recover strict radial monotonicity for the biharmonic equation in the unit ball of $ \mathbb{R}^{4} $.</p></abstract>

无穷Laplace方程的超定边值问题

无穷Laplace方程的超定边值问题

On the Radial Symmetry Property for Harmonic Functions

Let $\Gamma$ be a closed smooth Jordan curve in the complex plane $\mathbb{C}$ ,G be a bounded domain in $\mathbb{C}$ with the boundary $\Gamma$ , and let $\overline{G}=G\cup\Gamma$ . We study functions that are continuous in $\mathbb{C}\setminus G$ and harmonic in $\mathbb{C}\setminus\overline{G}$ that grow more slowly than the function $|z|^2$ at $z\to\infty$ . It is shown that, if in the class of such functions there exists a solution to the overdetermined Neumann boundary value problem in which the function equals zero on $\Gamma$ and $\mu$ -almost everywhere on $\Gamma$ there exists and equals one the normal derivative of this function, then the domain G is a disk (Theorem 1). In this case, the solution is unique and it coincides up to a constant with the fundamental solution for the Laplace operator in $\mathbb{C}$ with a singularity in the center of the disk G. The proof of Theorem 1 is based on the application of the conformal mapping of the exterior of the unit disk onto the domain $\mathbb{C}\setminus \overline{G}$ . This mapping allows us to reduce the original problem for the domain $\mathbb{C}\setminus \overline{G}$ to overdetermined boundary value problem for the exterior of the disk in which the main difficulty is the heterogeneity of the boundary condition for the normal derivative. To study this condition some subtle results were required on the boundary properties of a function that performs the indicated conformal mapping as well as some properties of the Hardy classes Hp in the unit disk. Theorem 2 of the paper shows that in the general case the conditions in Theorem 1 cannot be relaxed. It states the existence of a bounded domain $G\subset\mathbb{C}$ different from a disk with a smooth Jordan boundary $\Gamma$ and functions $f_1,f_2,f_3\in C(\mathbb{C}\setminus G)$ harmonic in $\mathbb{C}\setminus\overline{G}$ for each of which exactly one of the conditions of Theorem 1 is not satisfied.

Free boundary minimal surfaces and overdetermined boundary value problems

In this paper we establish a connection between free boundary minimal surfaces in a ball in ℝ3 and free boundary cones arising in a one-phase problem.

Nontrivial solutions to Serrin's problem in annular domains

We construct nontrivial bounded, real analytic domains Ω⊆Rn of the form Ω0∖Ω‾1, bifurcating from annuli, which admit a positive solution to the overdetermined boundary value problem{−Δu=1,u>0 in Ω,u=0,∂νu=const on ∂Ω0,u=const,∂νu=const on ∂Ω1, where ν stands for the inner unit normal to ∂Ω. From results by Reichel [1] and later by Sirakov [2], it was known that the condition ∂νu≤0 on ∂Ω1 is sufficient for rigidity to hold, namely, the only domains which admit such a solution are annuli and solutions are radially symmetric. Our construction shows that the condition is also necessary. In addition, we show that the constructed domains are self-Cheeger.