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The Wiener test and potential estimates for quasilinear elliptic equations

On montre que pour p fixe, 1<p<∞, si n est une solution faible de ⊇•(|⊇u| P−2 ⊇u)=0 dans un domaine borne D⊆R n alors ⊇u est continue de Holder sur des sous-ensembles compacts de D

Regularity of the derivatives of solutions to certain degenerate elliptic equations

We present a bibliography on moving and free boundary problems for the heatdiffusion equation, particularly regarding the Stefan and related problems. It contains 5869 titles referring to 588 scientific Journals, 122 books, 88 symposia (having at least 3 contributions on the subject), 30 collections, 59 thesis and 247 technical reports. It tries to give a comprehensive account of the western existing mathematicalphysical-engineering literature on this research field. RESUMEN Se presenta una bibliografía sobre problemas de frontera móvil y libre para la ecuación del calor-difusión, en particular sobre el problema de Stefan y problemas relacionados. Contiene 5869 títulos distribuidos en 588 revistas científicas, 122 libros, 88 simposios (teniendo al menos 3 contribuciones en el tema), 30 colecciones, 59 tesis y 247 informes técnicos o prepublicaciones. Se da un informe amplio de la bibliografía matemática, física y de las ingenierías existente en occidente sobre este tema de investigación. Primary Mathematics Subject Classification Number (*): 35R35, 80A22 Secondary Mathematics Subject Classification Number (*): 35B40, 35C05, 35C15, 35Kxx, 35R30, 46N20, 49J20, 65Mxx, 65Nxx, 76R50, 76S05, 76T05, 93C20. (*) Following the 1991 Mathematics Subject Classification compiled by Mathematical Reviews and Zentralblatt fur Mathematik. Primary key words: Enthalpy formulation or method, Filtration, Free boundary problems, Freezing, Melting, Moving boundary problems, Mushy region, Phase-change problem, Solidification, Stefan problem. Secondary key words: Continuous mechanics, Diffusion process, Functional analysis, Heat conduction, Mathematical methods, Numerical methods, Partial differential equations, Variational inequalities, Weak solutions. Palabras claves primarias: Método o formulación en entalpía, Filtración, Problemas de frontera libre, Congelación, Derretimiento, Problemas de frontera móvil, Región pastosa, Problema de cambio de fase, Solidificación, Problema de Stefan. Palabras claves secundarias: Mecánica del continuo, Procesos difusivos, Análisis funcional, Conducción del calor, Métodos matemáticos, Métodos numéricos, Ecuaciones diferenciales a derivadas parciales, Inecuaciones variacionales, Soluciones débiles. El manuscrito fue recibido y aceptado en octubre de 1999. D.A. Tarzia, A bibliography on FBP. The Stefan problem, MAT Serie A, # 2 (2000). 3

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A bibliography on moving-free boundary problems for the heat-diffusion equation. The stefan and related problems

A class of multidimensional degenerate parabolic equations is considered: the two-phase Stefan problem and the porous medium equation are analyzed as models of singular parabolic equations; nonstat...

Approximation of Degenerate Parabolic Problems Using Numerical Integration

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Degenerate elliptic equations with measure data and nonlinear potentials

In this paper we are concerned with the behavior at the boundary of weak solutions of the Dirichlet problem for quasilinear elliptic equations of second order in an open set O ~ IR". The main result is that a bounded weak solution is continuous at a boundary point of f2 provided that the complement of f2 in a neighborhood of this point is sufficiently " thick" when measured by an appropriate capacity. In the case of Laplace's equation this condition reduces to that considered by WIENER [W1], [W2"]. The equations considered are of the general divergence structure type

A regularity condition at the boundary for solutions of quasilinear elliptic equations

: Consideration of the nonsteady filtration of an incompressible fluid in a homogeneous, partially saturated, porous medium leads to a degenerate nonlinear parabolic equation which is of elliptic type in the space variables in the region of degeneracy, i.e. in the saturated region. Results concerning the existence of weak solutions of such equations have been established. Here the local properties of weak solutions are studied. In particular it is shown that under the realistic assumption that the hydrostatic potential is bounded, the solution (moisture content) can be defined pointwise and is continuous. (Author).

Ronald Gariepy

Papers

A regularity condition at the boundary for solutions of quasilinear elliptic equations

Local behavior of solutions of an elliptic-parabolic equation