SUPERLINEAR ELLIPTIC PROBLEMS WITH SIGN CHANGING COEFFICIENTS

Via variational methods, we study multiplicity of solutions for the problem {-Delta u = lambda b(x)vertical bar u vertical bar(q-2)u + au + g(x, u) in Omega, u - 0 on partial derivative Omega, where a simple example for g( x, u) is |u|(p-2)u; here a, lambda are real parameters, 1 < q < 2 < p <= 2* and b(x) is a function in a suitable space L-sigma. We obtain a class of sign changing coefficients b(x) for which two non-negative solutions exist for any lambda > 0, and a total of five nontrivial solutions are obtained when lambda is small and a >= lambda(1). Note that this type of results are valid even in the critical case.

The factorization method for real elliptic problems

The Factorization Method localizes inclusions inside a body from measurements on its surface. Without a priori knowing the physical parameters inside the inclusions, the points belonging to them can be characterized using the range of an auxiliary operator. The method relies on a range characterization that relates the range of the auxiliary operator to the measurements and is only known for very particular applications. In this work we develop a general framework for the method by considering symmetric and coercive operators between abstract Hilbert spaces. We show that the important range characterization holds if the difference between the inclusions and the background medium satisfies a coerciveness condition which can immediately be translated into a condition on the coefficients of a given real elliptic problem. We demonstrate how several known applications of the Factorization Method are covered by our general results and deduce the range characterization for a new example in linear elasticity.

hp-dGFEM for Second-Order Elliptic Problems in Plyhedra I: Stability and Quasioptimality on Geometric Meshes

We introduce and analyze hp-version discontinuous Galerkin (dG) finite element methods for the numerical approximation of linear second-order elliptic boundary-value problems in three-dimensional polyhedral domains. To resolve possible corner-, edge- and corner-edge singularities, we consider hexahedral meshes that are geometrically and anisotropically refined toward the corresponding neighborhoods. Similarly, the local polynomial degrees are increased linearly and possibly anisotropically away from singularities. We design interior penalty hp-dG methods and prove that they are well-defined for problems with singular solutions and stable under the proposed hp-refinements. We establish (abstract) error bounds that will allow us to prove exponential rates of convergence in the second part of this work.

hp-dGFEM for Second-Order Elliptic Problems in Plyhedra II: Exponential Convergence

The goal of this paper is to establish exponential convergence of $hp$-version interior penalty (IP) discontinuous Galerkin (dG) finite element methods for the numerical approximation of linear second-order elliptic boundary-value problems with homogeneous Dirichlet boundary conditions and piecewise analytic data in three-dimensional polyhedral domains. More precisely, we shall analyze the convergence of the $hp$-IP dG methods considered in [D. Schötzau, C. Schwab, T. P. Wihler, SIAM J. Numer. Anal., 51 (2013), pp. 1610--1633] based on axiparallel $\sigma$-geometric anisotropic meshes and $\bm{s}$-linear anisotropic polynomial degree distributions.

hp-DGFEM for second-order mixed elliptic problems polyhedra

We prove exponential rates of convergence of hp-version discontinuous Galerkin (dG) interior penalty finite element methods for second-order elliptic problems with mixed Dirichlet-Neumann boundary conditions in axiparallel polyhedra. The dG discretizations are based on axiparallel, σ-geometric anisotropic meshes of mapped hexahedra and anisotropic polynomial degree distributions of μ-bounded variation. We consider piecewise analytic solutions which belong to a larger analytic class than those for the pure Dirichlet problem considered in [11, 12]. For such solutions, we establish the exponential convergence of a nonconforming dG interpolant given by local L 2 -projections on elements away from corners and edges, and by suitable local low-order quasi-interpolants on elements at corners and edges. Due to the appearance of non-homogeneous, weighted norms in the analytic regularity class, new arguments are introduced to bound the dG consistency errors in elements abutting on Neumann edges. The non-homogeneous norms also entail some crucial modifications of the stability and quasi-optimality proofs, as well as of the analysis for the anisotropic interpolation operators. The exponential convergence bounds for the dG interpolant constructed in this paper generalize the results of [11, 12] for the pure Dirichlet case.

Existence and multiplicity results for quasilinear elliptic equations

The goal of this paper is to study the multiplicity of positive solutions of a class of quasilinear elliptic equations. Based on the mountain pass theorems and sub-and supersolutions argument for p-Laplacian operators, under suitable conditions on nonlinearity f (x, s), we show the following problem: -Delta(p)u = lambda f(x,u) in Omega, u/(partial derivative Omega) = 0, where Omega is a bounded open subset of R-N, N >= 2, with smooth boundary, lambda is a positive parameter and Delta(p) is the p-Laplacian operator with p > 1, possesses at least two positive solutions for large lambda.

Asymptotic bifurcation results for quasilinear elliptic operators

We develop results for bifurcation from the principal eigenvalue for certain operators based on the p-Laplacian and containing a superlinear nonlinearity with a critical Sobolev exponent. The main result concerns an asymptotic estimate of the rate at which the solution branch departs from the eigenspace. The method can also be applied for nonpotential operators.

Homogenization of elliptic problems in thin domains with oscillatory boundaries

Los dominios finos, es decir, dominios sustancialmente más pequeños en alguna o varias de sus direcciones que en el resto, aparecen en muchos campos de la ciencia. Por ejemplo, dinámica de fluídos (lubricación, conducción de fluídos en tubos delgados, dinámica de oceanos...), mecánica de sólidos (barras delgadas, placas o cáscaras) o incluso en fisiología (circulación de la sangre). Así, el amplio número de posibles aplicaciones a situaciones reales ha hecho que la investigación de modelos de ecuaciones en derivadas parciales en dominios finos se convierta en un tema muy estudiado en los últimos años. Desde un punto de vista matemático, el estudio de las soluciones de una EDP en un dominio fino es un caso particular de la cuestión general relativa a cómo la variación de los dominios afecta al comportamiento de las soluciones de la EDP. En este marco, obtener la ecuación límite del modelo considerado, comparar la solución de la ecuación límite y las soluciones del problema en el dominio fino, analizar los coeficientes de la ecuación límite y comprender cómo la geometría del dominio afecta a la ecuación límite son algunos de los objetivos que deberían ser alcanzados. De hecho, es importante señalar que este tipo de cuestiones no sólo proporcionan importantes resultados teóricos sino que son muy relevantes desde el punto de vista de las aplicaciones. Por ejemplo, ser capaz de reducir el problema original a un problema mucho más sencillo, problema límite, que refleje las principales características del problema de partida puede ser muy útil para ingenieros y físicos...

Quasilinear elliptic systems with measure data

We study the existence of solutions of quasilinear elliptic systems involving $N$ equations and a measure on the right hand side, with the form $$\left\{\begin{array}{ll} -\sum_{i=1}^n \frac{\partial}{\partial x_i}\left(\sum\limits_{\beta=1}^{N}\sum\limits_{j=1}^{n}% a_{i,j}^{\alpha,\beta}\left( x,u\right)\frac{\partial}{\partial x_j}u^\beta\right)=\mu^\alpha& \mbox{ in }\Omega ,\\ u=0 & \mbox{ on }\partial\Omega, \end{array}\right.$$ where $\alpha\in\{1,\dots,N\}$ is the equation index, $\Omega$ is an open bounded subset of $\mathbb{R}^{n}$, $u:\Omega\rightarrow\mathbb{R}^{N}$ and $\mu$ is a finite Randon measure on $\mathbb{R}^{n}$ with values into $\mathbb{R}^{N}$. Existence of a solution is proved for two different sets of assumptions on $A$. Examples are provided that satisfy our conditions, but do not satisfy conditions required on previous works on this matter.