Existence and summability of solutions to nonlinear X-elliptic equations with measurable coefficients

Existence of weak solutions of a nonlinear parabolic equation in a noncylindrical domain

In this paper, we study the eigenvalue problem of fourth-order elliptic operators in divergence form with weight on compact Riemannian manifolds with boundary (possibly empty) and get two types of general inequalities for them. By using these general inequalities, we can obtain some universal inequalities for eigenvalues of fourth-order elliptic operators in divergence form with weight on bounded domains in complete Riemannian manifolds. Furthermore, we also obtain some estimates for lower order eigenvalues of fourth-order elliptic operators in divergence form with weight on compact submanifolds in Euclidean spaces.

Nonlinear parabolic Cauchy–Dirichlet problems with lower order term of natural growth in the gradient and irregular source/initial data

Given a parabolic cylinder Q = (0,T) ! ! , where ! R N is a bounded domain, we prove new properties of solutions of ut # pu = µ in Q with Dirichlet boundary conditions, where µ is a finite Radon measure in Q. We first prove a priori estimates on the p-parabolic capacity of level sets of u. We then show that diffuse measures (i.e., mea- sures which do not charge sets of zero parabolic p-capacity) can be strongly approximated by the measures µk = (Tk(u))t# p(Tk(u)),andweintroduceanewnotionofrenormalizedsolutionbasedonthisproperty. We finally apply our new approach to prove the existence of solutions of ut # pu + h(u) = µ in Q, for any function h such that h(s)s $ 0 and for any diffuse measure µ; when h is nondecreasing, we also prove uniqueness in the renormalized formulation. Extensions are given to the case of more general nonlinear operators in divergence form.

Mixed finite element methods for treating the Dirichlet problem for fully nonlinear second-order elliptic operators in divergence form are extended to cover the three-dimensional case. Existence and uniqueness of the approximation are proved, and optimal error estimates in L2 are demonstrated for both the scalar and vector functions approximated by the method. Error estimates for the pressure variable are also derived in Lq; the result is optimal in order for 2 ≤ q ≤ 6 and less than optimal for 6 < q ≤ + ∞. Newton's method can be used to solve the nonlinear algebraic equations. © 1996 John Wiley & Sons, Inc.

Positive solutions of some eigenvalue problems in the theory of variational inequalities

Pohozaev identity plays a very important role in proving the existence and nonexistence results for the nonlinear elliptic partial differential equations. In this review, we aim to introduce Pohozaev identity and its applications in some classical elliptic partial differential equations. First, we introduce Pohozaev identities for several typical elliptic equations on bounded domains or unbounded domains, and the necessary conditions for the existence of solutions, by which we get some existence and nonexistence results of the solutions. Next, we will show a local Pohozaev identity and its application in proving the compactness of the sequences of approximate solutions, by which we obtain infinitely many solutions of some typical nonlinear elliptic equations. At last, we also introduce some local uniqueness results of peak solutions for some nonlinear elliptic partial differential equations, which can also be obtained by using the local Pohozaev identity. Using this type of uniqueness result, we can prove that the peak solutions can keep the symmetry that the equations have.

In this paper, we study eigenvalues of elliptic operators in divergence form on compact Riemannian manifolds with boundary (possibly empty) and obtain a general inequality for them. By using this inequality, we prove universal inequalities for eigenvalues of elliptic operators in divergence form on compact domains of complete submanifolds in a Euclidean space, and of complete manifolds admitting special functions which include the Hadamard manifolds with Ricci curvature bounded below, a class of warped product manifolds, the product of Euclidean spaces with any complete manifold and manifolds admitting eigenmaps to a sphere.

We establish a method for giving lower bounds for the fundamental tone of elliptic operators in divergence form in terms of the divergence of vector fields. We then apply this method to the Lr operator associated to immersed hypersurfaces with locally bounded (r + 1)-th mean curvature Hr + 1 of the space forms Nn+ 1(c) of constant sectional curvature c. As a corollary we give lower bounds for the extrinsic radius of closed hypersurfaces of Nn+ 1(c) with Hr + 1 > 0 in terms of the r-th and (r + 1)-th mean curvatures. Finally we observe that bounds for the Laplace eigenvalues essentially bound the eigenvalues of a self-adjoint elliptic differential operator in divergence form. This allows us to show that Cheeger's constant gives a lower bounds for the first nonzero Lr-eigenvalue of a closed hypersurface of Nn+ 1(c).

Existence, uniqueness, and stability of nonlinear evolution equations

In this paper we prove the existence of solutions of nonlinear equations of the type-div(a(x, u, Du)+H(x, u, Du)=f, wherea andH are Caratheodory functions andf is a bounded Radon measure. We remark that the operator can be not coercive. We give also some regularity results.

In this paper, exploiting variational methods, the existence of three weak solutions for a class of elliptic equations involving a general operator in divergence form and with Dirichlet boundary condition is investigated. Several special cases are analyzed. In conclusion, for completeness, a concrete example of an application is presented by finding the existence of three nontrivial weak solutions for an uniformly elliptic second-order problem on a bounded Euclidean domain.

Some isoperimetric problems for the principal eigenvalues of second-order elliptic operators in divergence form

We consider a second‐order differential operator A(x)=−∑∂iaij(x)∂j+ ∑∂j(bj(x)·)+c(x) on ℝd, on a bounded domain D with Dirichlet boundary conditions on ∂D, under mild assumptions on the coefficients of the diffusion tensor aij. The object is to construct monotone numerical schemes to approximate the solution of the problem A(x)u(x)=µ(x), x∈D, where µ is a positive Radon measure. We start by briefly mentioning questions of existence and uniqueness introducing function spaces needed to prove convergence results. Then, we define non‐standard stencils on grid‐knots that lead to extended discretization schemes by matrices possessing compartmental structure. We proceed to discretization of elliptic operators, starting with constant diffusion tensor and ending with operators in divergence form. Finally, we discuss W‐convergence in detail, and mention convergence in C and L1 spaces. We conclude by a numerical example illustrating the schemes and convergence results. Copyright © 2008 John Wiley &amp; Sons, Ltd.

In this paper, we study eigenvalue of linear fourth order elliptic operators in divergence form with Dirichlet boundary condition on a bounded domain in a compact Riemannian manifolds with boundary (possibly empty) and provide a general inequality for them. As an application, by using this inequality, we study eigenvalues of this operator on compact domains of complete submanifolds in a Euclidean space.