Dirichlet problem for parabolic systems with Dini continuous coefficients on the plane

The Dirichlet problem for a one-dimensional (with respect to x) second-order parabolic system with Dini continuous coefficients is considered in an x-semibounded domain with a nonsmooth lateral boundary from the Dini–Hölder class. The classical solvability of the problem is proved by applying the method of boundary integral equations. The only condition imposed on the right-hand side of the boundary condition is that it has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied.     

A domain decomposition method for linear exterior boundary value problems

In this paper, we present a domain decomposition method, based on the general theory of Steklov-Poincaré operators, for a class of linear exterior boundary value problems arising in potential theory and heat conductivity. We first use a Dirichlet-to-Neumann mapping, derived from boundary integral equation methods, to transform the exterior problem into an equivalent mixed boundary value problem on a bounded domain. This domain is decomposed into a finite number of annular subregions, and the Dirichlet data on the interfaces is introduced as the unknown of the associated Steklov-Poincaré problem. This problem is solved with the Richardson method by introducing a Dirichlet-Robin-type preconditioner, which yields an iteration-by-subdomains algorithm well suited for parallel computations. The corresponding analysis for the finite element approximations and some numerical experiments are also provided.     

Model oblique derivative problem for the heat equation with a discontinuous boundary function

The oblique derivative problem for the heat equation is considered in a model formulation with a boundary function that can be discontinuous and with the boundary condition understood as the limit in the normal direction almost everywhere on the lateral boundary of the domain. An example is given showing that the solution is not unique in this formulation. A solution is sought in the parabolic Zygmund space H ₁, which is an analogue of the parabolic Hölder space for an integer smoothness exponent. A subspace of H ₁ is introduced in which the existence and uniqueness of the solution is proved under suitable assumptions about the data of the problem.     

The Dirichlet problem for the minimal hypersurface equation on arbitrary domains of a Riemannian manifold

We show that the Dirichlet problem for the minimal hypersurface equation defined on arbitrary C ² bounded domain Ω of an arbitrary complete Riemannian manifold M is solvable if the oscillation of the boundary data is bounded by a function \({\mathcal{C}}\) that is explicitely given and that depends only on the first and second derivatives of the boundary data as well as the second fundamental form of the boundary \({\partial\Omega}\) and the Ricci curvature of the ambient space M. This result extends Theorem 2 of Jenkins-Serrin (J Reine Angew Math 229:170–187,1968) about the solvability of the Dirichlet problem for the minimal hypersurface equation defined on bounded domains of the Euclidean space. We deduce that the Dirichlet problem for the minimal hypersurface equation is solvable for any continuous boundary data on a mean convex domain. We also show existence and uniqueness of the Dirichlet problem with boundary data at infinity—exterior Dirichlet problem—on Hadamard manifolds.     

A symmetry problem with applications to finance

In this paper we will consider an overdetermined problem in an unknown ring-shaped domain, and we prove that the domain has to be spherical ring. We will apply this problem to an \(n\) -dimensional toy model in the mean field game theory introduced by Lasry-Lions too. We show that if in Lasry-Lions model, the additional extra data ”the boundary is a level set” is assumed, then the region, where the solution is harmonic, has to have spherical symmetry.     

The dirichlet problem with generalized functions as data

Elliptic boundary value problems with analytic functionals as data have been studied by Lions and Magenes. In this papér we relax their assumption that the boundary of the domain ω is an analytic surface; we assume only that ω equals the interior of its closure. In this case we obtain results for the Dirichlet problem for second order equations that are analogous to theirs: There is a quasi-analytic class of functions on⁰ω with a natural topology such that the Dirichlet problem is well posed the data belongs to the dual of this space. The author acknowledges with gratitude the support he received from the National Science Foundation through a graduate fellowship and NSFGP 6761. Entrata in redazione il 31 gennaio 1969     

A study of the stability of solutions of linear parabolic equations with nearly constant coefficients and small diffusion

We study the stability of the solutions of boundary value problems for a certain class of Petrovskii-parabolic systems with sufficiently small diffusion coefficients. The dimension of the set of critical cases in the stability problem turns out to be infinite. We develop an efficient algorithm for studying stability. As examples we consider parabolic boundary value problems with delay and rapidly oscillating coefficients, the problem of parametric resonance under a double-frequency perturbation, and problems with variable leading terms and variable domain of definition. Bibliography: 21 titles.Translated fromTrudy Seminara imeni I. G. Petrovskogo, No. 15, pp. 128–155, 1991.     

Optimal Shape Problems for Eigenvalues

ABSTRACT

We consider the first Dirichlet eigenvalue for nonhomogeneous membranes. For given volume we want to find the domain which minimizes this eigenvalue. The problem is formulated as a variational free boundary problem. The optimal domain is characterized as the support of the first eigenfunction. We prove enough regularity for the eigenfunction to conclude that the optimal domain has finite parameter. Finally an overdetermined boundary value problem on the regular part of the free boundary is given.     

A Riemann-Hilbert Problem for the Moisil-Teodorescu System

In a bounded domain with smooth boundary in ?³ we consider the stationary Maxwell equations for a function u with values in ?³ subject to a nonhomogeneous condition (u, v)_x = u₀ on the boundary, where v is a given vector field and u₀ a function on the boundary. We specify this problem within the framework of the Riemann-Hilbert boundary value problems for the Moisil-Teodorescu system. This latter is proved to satisfy the Shapiro-Lopaniskij condition if an only if the vector v is at no point tangent to the boundary. The Riemann-Hilbert problem for the Moisil-Teodorescu system fails to possess an adjoint boundary value problem with respect to the Green formula, which satisfies the Shapiro-Lopatinskij condition. We develop the construction of Green formula to get a proper concept of adjoint boundary value problem.     

The Neumann problem for the planar Stokes system

The Neumann problem for the Stokes system is studied on bounded and unbounded domains with Ljapunov boundary (i.e. of class ${{\mathcal C}^{1,\alpha }}$ ) in the plane. We construct a solution of this problem in the form of appropriate potentials and reduce the problem to an integral equation systems on the boundary of the domain. We determine a necessary and sufficient condition for the solvability of the problem. Then we study the direct integral equation method and prove that a solution of the corresponding integral equation can be obtained by the successive approximation.     

Grid approximation of singularly perturbed parabolic equations with piecewise continuous initial-boundary conditions

The Dirichlet problem is considered for a singularly perturbed parabolic reaction-diffusion equation with piecewise continuous initial-boundary conditions in a rectangular domain. The highest derivative in the equation is multiplied by a parameter ? ², ? ε (0, 1]. For small values of the parameter ?, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristic of the limit equation passing through the point of discontinuity of the initial function, there arise a boundary layer and an interior layer (of characteristic width ?), respectively, which have bounded smoothness for fixed values of the parameter ?. Using the method of additive splitting of singularities (generated by discontinuities of the boundary function and its low-order derivatives), as well as the method of condensing grids (piecewise uniform grids condensing in a neighborhood of boundary layers), we construct and investigate special difference schemes that converge ?-uniformly with the second order of accuracy in x and the first order of accuracy in t, up to logarithmic factors.     

Boundary-value problems for an elliptic equation with complex coefficients and a certain moment problem

Elliptic systems of two second-order equations, which can be written as a single equation with complex coefficients and a homogeneous operator, are studied. The necessary and sufficient conditions for the connection of traces of a solution are obtained for an arbitrary bounded domain with a smooth boundary. These conditions are formulated in the form of a certain moment problem on the boundary of a domain; they are applied to the study of boundary-value problems. In particular, it is shown that the Dirichlet problem and the Neumann problem are solvable only together. In the case where the domain is a disk, the indicated moment problem is solved together with the Dirichlet problem and the Neumann problem. The third boundary-value problem in a disk is also investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 11, pp. 1476–1483, November, 1993.     

Nonlinear artificial boundary conditions with pointwise error estimates for the exterior three dimensional Navier–Stokes problem

On a three–dimensional exterior domain Ω we consider the Dirichlet problem for the stationary Navier–Stokes system. We construct an approximation problem on the domain Ω_R, which is the intersection of Ω with a sufficiently large ball, while we create nonlinear, but local artificial boundary conditions on the truncation boundary. We prove existence and uniqueness of the solutions to the approximating problem together with asymptotically precise pointwise error estimates as R tends to infinity.     

Existence and semiclassical limit for the quantum drift-diffusion model

In this paper, we construct a weak solution to the unipolar quantum drift-diffusion model in a bounded domain with Neumann boundary conditions, thereby extending an existence result in Gianazza et al. (Arch Rational Mech Anal 194:133–220, 2009) to the case where the boundary of the domain is only Lipschitz. We also obtain the limit of the solution as the scaled Planck constant \(\varepsilon \) in the problem goes to \(0\) .     

A mixed boundary value problem for a singularly perturbed reaction-diffusion equation in an L-shaped domain

A mixed boundary value problem for a singularly perturbed reaction-diffusion equation in an L-shaped domain is considered for when the solution has singularities at the corners of the domain. The densification of the Shishkin mesh near the inner corner where different boundary conditions meet is such that the solution obtained by the classical five-point difference scheme converges to the solution of the initial problem in the mesh norm L _?? ^h uniformly with respect to the small parameter with almost second order, i.e., as a smooth solution. Numerical analysis confirms the theoretical result.     

Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems

Summary This paper considers the optimal quadratic cost problem (regulator problem) for a class of abstract differential equations with unbounded operators which, under the same unified framework, model in particular «concrete» boundary control problems for partial differential equations defined on a bounded open domain of any dimension, including: second order hyperbolic scalar equations with control in the Dirichlet or in the Neumann boundary conditions; first order hyperbolic systems with boundary control; and Euler-Bernoulli (plate) equations with (for instance) control(s) in the Dirichlet and/or Neumann boundary conditions. The observation operator in the quadratic cost functional is assumed to be non-smoothing (in particular, it may be the identity operator), a case which introduces technical difficulties due to the low regularity of the solutions. The paper studies existence and uniqueness of the resulting algebraic (operator) Riccati equation, as well as the relationship between exact controllability and the property that the Riccati operator be an isomorphism, a distinctive feature of the dynamics in question (emphatically not true for, say, parabolic boundary control problems). This isomorphism allows one to introduce a «dual» Riccati equation, corresponding to a «dual» optimal control problem. Properties between the original and the «dual» problem are also investigated.Research partially supported by the National Science Foundation under Grant NSF-DMS-8301668 and by the Air Force Office of Scientific Research under Grant AFOSR-84-0365.     

Nonlinear fourth-order elliptic equations with nonlocal boundary conditions

This paper is concerned with a class of fourth-order nonlinear elliptic equations with nonlocal boundary conditions, including a multi-point boundary condition in a bounded domain of Rn. Also considered is a second-order elliptic equation with nonlocal boundary condition, and the usual multi-point boundary problem in ordinary differential equations. The aim of the paper is to show the existence of maximal and minimal solutions, the uniqueness of a positive solution, and the method of construction for these solutions. Our approach to the above problems is by the method of upper and lower solutions and its associated monotone iterations. The monotone iterative schemes can be developed into computational algorithms for numerical solutions of the problem by either the finite difference method or the finite element method.     

Well-posedness of the Poincar�� problem for a multidimensional hyperbolic equation with the Chaplygin operator

In this paper, the unique solvability of the Poincaré problem for multidimensional hyperbolic equation with the Chaplygin operator in the domain with a deviation from the characteristic is proved. In the theory of partial differential equations of the hyperbolic type, boundary-value problems with the data on the whole boundary of the domain serve as an example of problems that are not well posed [3].     

Viscosity solutions of a class of degenerate quasilinear parabolic equations with Dirichlet boundary condition

This paper is concerned with viscosity solutions for a class of degenerate quasilinear parabolic equations in a bounded domain with homogeneous Dirichlet boundary condition. The equation under consideration arises from a number of practical model problems including reaction–diffusion processes in a porous medium. The degeneracy of the problem appears on the boundary and possibly in the interior of the domain. The goal of this paper is to establish some comparison properties between viscosity upper and lower solutions and to show the existence of a continuous viscosity solution between them. An application of the above results is given to a porous-medium type of reaction–diffusion model which demonstrates some distinctive properties of the solution when compared with the corresponding semilinear problem.