Dirichlet problem for a nonlocal wave equation

We study the Dirichlet problem for a nonlocal wave equation in a rectangular domain. We prove the existence and uniqueness of a solution of the problem and show that determining whether the solution is unique can be reduced to determining whether a function of Mittag-Leffler type has real zeros. The obtained uniqueness condition turns into the uniqueness condition for the solution of the Dirichlet problem for the wave equation as the order of the fractional derivative in the equation tends to 2.     

A numerical method for solving the Dirichlet problem for the wave equation

In this paper we present a numerical method for solving the Dirichlet problem for a two-dimensional wave equation. We analyze the ill-posedness of the problem and construct a regularization algorithm. Using the Fourier series expansion with respect to one variable, we reduce the problem to a sequence of Dirichlet problems for one-dimensional wave equations. The first stage of regularization consists in selecting a finite number of problems from this sequence. Each of the selected Dirichlet problems is formulated as an inverse problem Aq = f with respect to a direct (well-posed) problem. We derive formulas for singular values of the operator A in the case of constant coefficients and analyze their behavior to judge the degree of ill-posedness of the corresponding problem. The problem Aq = f on a uniform grid is reduced to a system of linear algebraic equations A _ll q = F. Using the singular value decomposition, we find singular values of the matrix A _ll and develop a numerical algorithm for constructing the r-solution of the original problem. This algorithm was tested on a discrete problem with relatively small number of grid nodes. To improve the calculated r-solution, we applied optimization but observed no noticeable changes. The results of computational experiments are illustrated.     

The exterior Dirichlet problem for the Helmholtz equation

The exterior Dirichlet problem for the reduced wave equation is reformulated as a new integral equation. It is shown that the normal derivative of the total field may be expressed as a Neumann series in terms of the known incident field. The convergence of the infinite series is established for arbitrary smooth surfaces and for small values of the wave number. An example is given that illustrates the method.     

Dirichlet problem for the Stokes equation

The paper presents some coercive a priori estimates of the solution of the Dirichlet problem for the linear Stokes equation relating vorticity and the stream function of an axially symmetric flow of an incompressible fluid. This equation degenerates on the axis of symmetry. The method used to obtain the estimates is based on a differential substitution transforming the Stokes equation into the Laplace equation and on the subsequent transition from cylindrical to Cartesian coordinates in three-dimensional space.     

The Dirichlet problem for the Bellman equation at resonance

We generalize the Donsker-Varadhan minimax formula for the principal eigenvalue of a uniformly elliptic operator in nondivergence form to the first principal half-eigenvalue of a fully nonlinear operator which is concave (or convex) and positively homogeneous. Examples of such operators include the Bellman operator and the Pucci extremal operators. In the case that the two principal half-eigenvalues are not equal, we show that the measures which achieve the minimum in this formula provide a partial characterization of the solvability of the corresponding Dirichlet problem at resonance.     

The Dirichlet boundary problem for a nonlocal Cahn-Hilliard equation

We study the existence, uniqueness and continuous dependence on initial data of the solution for a nonlocal Cahn-Hilliard equation with Dirichlet boundary condition on a bounded domain. Under a nondegeneracy assumption the solutions are classical but when this is relaxed, the equation is satisfied in a weak sense. Also we prove that there exists a global attractor in some metric space.     

Dirichlet problem for the Boussinesq–Love equation

We study the cases of unique solvability of the Dirichlet problem for the Boussinesq–Love equation.     

The Dirichlet problem for the minimal surface equation in non-regular domains

Summary In this paper we study the Dirichlet problem for the minimal surface equation in a open set Ω without any assumption about the regularity of ϖΩ. We prove an existence theorem using only the pseudoconvexity of Ω.

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Riassunto In questo lavoro studiamo il problema di Dirichlet per l'equazione delle superfici minime in un aperto Ω diR ⁿ sulla cui frontiera non si fa nessuna ipotesi di regolarità. Si ottiene un teorema di esistenza usando la sola pseudoconvessità di Ω.

     

The Dirichlet problem associated to the relativistic heat equation

We prove existence and uniqueness of entropy solutions for the nonhomogeneous Dirichlet problem associated to the relativistic heat equation.     

The Dirichlet problem for telegraph equation in a rectangular domain

We investigate the Dirichlet problem for the telegraph equation in a rectangular domain. We establish a criterion of uniqueness of solution to the problem. The solution is constructed as the sum of an orthogonal series. In substantiation of convergence of the series, the problem of small denominators occurs. In connection with this, we establish estimates ensuring separation from zero of denominators with the corresponding asymptotics which allow us to prove the existence of a regular solution and prove its stability under small perturbations of boundary functions.