Blowing up boundary conditions for a nonlocal nonlinear diffusion equation in several space dimensions

We analyze boundary value problems prescribing Dirichlet or Neumann boundary conditions for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation in a bounded smooth domain Ω∈R^N with N≥1. First, we prove existence and uniqueness of solutions and the validity of a comparison principle for these problems. Next, we impose boundary data that blow up in finite time and study the behavior of the solutions.     

Radial solutions of Dirichlet problems with concave-convex nonlinearities

We prove the existence of a double infinite sequence of radial solutions for a Dirichlet concave-convex problem associated with an elliptic equation in a ball of Rⁿ. We are interested in relaxing the classical positivity condition on the weights, by allowing the weights to vanish. The idea is to develop a topological method and to use the concept of rotation number. The solutions are characterized by their nodal properties.     

A Note on Strong Solutions to the Stokes System

We give an alternative and quite simple proof of existence of W ^2,q -W ^1,q -strong solutions for the Stokes system, endowed with Dirichlet boundary conditions in a bounded smooth domain.     

On the asymptotic behavior of solutions of quasilinear elliptic equations

The asymptotic behavior of solutions of second-order quasilinear elliptic and nonhyperbolic partial differential equations defined on unbounded domains inR ⁿ contained in\(\{ x_1 ,...,x_n :\left| {x_n } \right|< \lambda \sqrt {x_1^2 + ...x_{n - 1}^2 } \) for certain sublinear functions λ is investigated when such solutions satisfy Dirichlet boundary conditions and the Dirichlet boundary data has appropriate asymptotic behavior at infinity. We prove Phragmèn-Lindelöf theorems for large classes of nonhyperbolic operators, without «lower order terms”, including uniformly elliptic operators and operators with well-definedgenre, using special barrier functions which are constructed by considering an operator associated to our original operator. We also estimate the rate at which a solution converges to its limiting function at infinity in terms of properties of the top order coefficienta _nn of the operator and the rate at which the boundary values converge to their limiting function; these results are proven using appropriate barrier functions which depend on the behavior of the coefficients of the operator and the rate of convergence of boundary values.     

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.     

Existence and regularity results for relaxed Dirichlet problems with measure data

We study the following relaxed Dirichlet problem $$\left\{ \begin{gathered} Lu + \mu u = vin\Omega , \hfill \\ u = 0on\partial \Omega , \hfill \\ \end{gathered} \right.$$ where Ω is a bounded open subset ofR ^N,Lu=?div(A?u) is an elliptic operator, μ is a positive Borel measure on Ω not charging polar sets, and v is a measure with bounded variation on Ω. We give a definition of solution for such a problem, and then prove existence and regularity results. As a consequence, the Green function for relaxed Dirichlet problems can be defined, and some of its properties are proved, including the standard representation formula for solutions.     

GLOBAL EXISTENCE, UNIFORM DECAY AND EXPONENTIAL GROWTH FOR A CLASS OF SEMI-LINEAR WAVE EQUATION WITH STRONG DAMPING

In this paper, we consider the nonlinearly damped semi-linear wave equation associated with initial and Dirichlet boundary conditions. We prove the existence of a local weak solution and introduce a family of potential wells and discuss the invariants and vacuum isolating behavior of solutions. Furthermore, we prove the global existence of solutions in both cases which are polynomial and exponential decay in the energy space respectively, and the asymptotic behavior of solutions for the cases of potential well family with 0相似文献   

Criteria of solvability for multidimensional Riccati equations

We study the solvability problem for the multidimensional Riccati equation ??u=|?u|^q+ω, whereq>1 and ω is an arbitrary nonnegative function (or measure). We also discuss connections with the classical problem of the existence of positive solutions for the Schrödinger equation ?Δu?ωu=0 with nonnegative potential ω. We establish explicit criteria for the existence of global solutions onR ⁿ in terms involving geometric (capacity) estimates or pointwise behavior of Riesz potentials, together with sharp pointwise estimates of solutions and their gradients. We also consider the corresponding nonlinear Dirichlet problem on a bounded domain, as well as more general equations of the type?Lu=f(x, u, ?u)+ω where , andL is a uniformly elliptic operator.     

Uniform decay rates for the energy of weakly damped defocusing semilinear Schrödinger equations with inhomogeneous Dirichlet boundary control

In this paper, we study the open loop stabilization as well as the existence and regularity of solutions of the weakly damped defocusing semilinear Schrödinger equation with an inhomogeneous Dirichlet boundary control. First of all, we prove the global existence of weak solutions at the H¹-energy level together with the stabilization in the same sense. It is then deduced that the decay rate of the boundary data controls the decay rate of the solutions up to an exponential rate. Secondly, we prove some regularity and stabilization results for the strong solutions in H²-sense. The proof uses the direct multiplier method combined with monotonicity and compactness techniques. The result for weak solutions is strong in the sense that it is independent of the dimension of the domain, the power of the nonlinearity, and the smallness of the initial data. However, the regularity and stabilization of strong solutions are obtained only in low dimensions with small initial and boundary data.     

Elliptic problems in generalized Orlicz-Musielak spaces

We consider a strongly nonlinear monotone elliptic problem in generalized Orlicz-Musielak spaces. We assume neither a ??₂ nor ?₂-condition for an inhomogeneous and anisotropic N-function but assume it to be log-H?lder continuous with respect to x. We show the existence of weak solutions to the zero Dirichlet boundary value problem. Within the proof the L ^??-truncation method is coupled with a special version of the Minty-Browder trick for non-reflexive and non-separable Banach spaces.     

Logistic Dirichlet problems with discontinuous coefficients

This paper is devoted to the study of the existence of positive solutions of semilinear Dirichlet eigenvalue problems for diffusive logistic equations with discontinuous coefficients which model population dynamics in environments with spatial heterogeneity. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of singular integral operators. Moreover, we make use of an L^p variant of an estimate for the Green operator of the Dirichlet problem introduced in the study of Feller semigroups.     

Explosion of Jump-Type Symmetric Dirichlet Forms on ℝ d

We give a sufficient condition for a class of jump-type symmetric Dirichlet forms on ? ^d to be conservative in terms of the jump kernel and the associated measure. Our condition allows the coefficients dominating big jumps to be unbounded. We derive the conservativeness for Dirichlet forms related to symmetric stable processes. We also show that our criterion is sharp by using time changed Dirichlet forms. We finally remark that our approach is applicable to jump-diffusion type symmetric Dirichlet forms on ? ^d .     

Robin Approximation of Dirichlet Boundary Value Problems

This paper describes the rate of convergence of solutions of Robin boundary value problems of an elliptic equation to the solution of a Dirichlet problem as a boundary parameter decreases to zero. The results are found using representations for solutions of the equations in terms of Steklov eigenfunctions. Particular interest is in the case where the Dirichlet data is only in L²(?Ω,dσ). Various approximation bounds are obtained and the rate of convergence of the Robin approximations in the H¹ and L² norms are shown to have convergence rates that depend on the regularity of the Dirichlet data.     

Oscillatory radial solutions for subcritical biharmonic equations

It is well known that the biharmonic equation Δ²u=u|u|^p−1 with p∈(1,∞) has positive solutions on Rn if and only if the growth of the nonlinearity is critical or supercritical. We close a gap in the existing literature by proving the existence and uniqueness, up to scaling and symmetry, of oscillatory radial solutions on Rn in the subcritical case. Analyzing the nodal properties of these solutions, we also obtain precise information about sign-changing large radial solutions and radial solutions of the Dirichlet problem on a ball.     

A nonlocal nonlinear diffusion equation in higher space dimensions

We study the initial-value problem for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation, in the whole RN, N?1, or in a bounded smooth domain with Neumann or Dirichlet boundary conditions. First, we prove the existence, uniqueness and the validity of a comparison principle for solutions of these problems. In RN we show that if initial data is bounded and compactly supported, then the solutions is compactly supported for all positive time t, this implies the existence of a free boundary. Concerning the Neumann problem, we prove that the asymptotic behavior of the solutions as t→∞, they converge to the mean value of the initial data. For the Dirichlet problem we prove that the asymptotic behavior of the solutions as t→∞, they converge to zero.     

Uniqueness and structure of solutions to the Dirichlet problem for an elliptic system

In this paper, we consider the Dirichlet problem for an elliptic system on a ball in R². By investigating the properties for the corresponding linearized equations of solutions, and adopting the Pohozaev identity and Implicit Function Theorem, we show the uniqueness and the structure of solutions.     

On W-solvability for special vectorial Hamilton-Jacobi systems

We study the solvability of special vectorial Hamilton-Jacobi systems of the form F(Du(x))=0 in a Sobolev space. In this paper we establish the general existence theorems for certain Dirichlet problems using suitable approximation schemes called W^1,p-reduction principles that generalize the similar reduction principle for Lipschitz solutions. Our approach, to a large extent, unifies the existing methods for the existence results of the special Hamilton-Jacobi systems under study. The method relies on a new Baire's category argument concerning the residual continuity of a Baire-one function. Some sufficient conditions for W^1,p-reduction are also given along with certain generalization of some known results and a specific application to the boundary value problem for special weakly quasiregular mappings.     

Domain perturbations and estimates for the solutions of second order elliptic equations

We study the dependence of the variational solution of the inhomogeneous Dirichlet problem for a second order elliptic equation with respect to perturbations of the domain. We prove optimal L² and energy estimates for the difference of two solutions in two open sets in terms of the “distance” between them and suitable geometrical parameters which are related to the regularity of their boundaries. We derive such estimates when at least one of the involved sets is uniformly Lipschitz: due to the connection of this problem with the regularity properties of the solutions in the L² family of Sobolev–Besov spaces, the Lipschitz class is the reasonably weakest one compatible with the optimal estimates.     

Dirichlet problem on locally finite graphs

In this paper, we study the existence and uniqueness of solutions to the vertex-weighted Dirichlet problem on locally finite graphs. Let B be a subset of the vertices of a graph G. The Dirichlet problem is to find a function whose discrete Laplacian on G?B and its values on B are given. Each infinite connected component of G?B is called an end of G relative to B. If there are no ends, then there is a unique solution to the Dirichlet problem. Such a solution can be obtained as a limit of an averaging process or as a minimizer of a certain functional or as a limit-solution of the heat equation on the graph. On the other hand, we show that if G is a locally finite graph with l ends, then the set of solutions of any Dirichlet problem, if non-empty, is at least l-dimensional.