# A priori properties of solutions of nonlinear equations with degenerate coercivity and L 1-data

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We prove an assertion about the increase of a solution, weak in the sense of Trudinger, to the Dirichlet problem for m-Hessian equations with the righthand side in L q, q > n( n + 1)/(2 m). We estimate the ratio between the increment of the solution along the normal and the distance to the...

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The work deals with the Dirichlet problem for elliptic equations with nonhomogeneous anisotropic degeneracy in a possibly unbounded domain of multidimensional Euclidean space. The existence of weak solutions is proved. Some conditions are established connecting the character of nonlinearity of...

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We study a class of shape optimization problems for semi-linear elliptic equations with Dirichlet boundary conditions in smooth domains in R[sup 2]. A part of the boundary of the domain is variable as the graph of a smooth function. The problem is equivalently reformulated on a fixed domain....

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We construct a sequence of solutions of the exterior Helmholtz equation such that their restrictions form an orthonormal basis on a given surface. The dependence of the coefficients of these functions on the coefficients of the surface are given by an explicit algebraic formula. In the same way,...

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We study the behavior of eigenvalues and eigenfunctions of the Dirichlet problem for nonlinear elliptic second-order equations in domains with fine-grain boundary.

- PRINCIPAL EIGENVALUE AND MAXIMUM PRINCIPLE FOR SOME ELLIPTIC SYSTEMS DEFINED ON GENERAL DOMAINS WITH REFINED DIRICHLET BOUNDARY CONDITION. Alziary, Bénédicte; Fleckinger, Jacqueline; Lécureux, Marie-Hélène // Communications in Mathematical Analysis;2009, Vol. 7 Issue 2, p1
We prove the existence of a principal eigenvalue and we derive a "Refined Maximum Principle" for an elliptic system LU = LMU +F defined on an irregular bounded domain in RN with Refined Dirichlet Boundary Condition; here L is a diagonal matrix of uniformly elliptic operators Li, 1 â©½ i...

- EXISTENCE RESULTS FOR STRONGLY INDEFINITE ELLIPTIC SYSTEMS. JIANFU YANG; YING YE; XIAOHUI YU // Electronic Journal of Differential Equations;2008, Vol. 2008, Special section p1
In this paper, we show the existence of solutions for the strongly indefinite elliptic system -Î”u = Î»u + f(x, v) in Î©, -Î”v = Î»v + g(x, u) in Î©, u = v = 0, on âˆ‚Î©, where Î© is a bounded domain in â„N (N â‰¥ 3) with smooth boundary, Î»k0 < Î» < Î»k0+1,...

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When an unbounded domain is inside a slab, existence of a positive solution is proved for the Dirichlet problem of a class of semilinear elliptic equations similar to the singular Emden-Fowler equation. The proof is based on a super and sub-solution method. A super solution is constructed by...

- Multiple solutions to non-convex variational problems with implications for phase transitions and numerical computation. GAO, D. Y.; OGDEN, R. W. // Quarterly Journal of Mechanics & Applied Mathematics;Nov2008, Vol. 61 Issue 4, p497
Non-convex variational/boundary-value problems are studied using a modified version of the Ericksen bar model in nonlinear elasticity. The strain-energy function is a general fourth-order polynomial in a suitable measure of strain that provides a convenient model for the study of, for example,...