The partial differential equations are a fundamental tool in the contemporary applied mathematics. Their importance is explained by the fact that various physical, chemical and other real processes are mathematically modeled through differential equations. On the other hand, the jump character of a grate part of the natural phenomena logically leads to their describing by problems with discontinuous data. This explains the rapid growth of the theory of PDE in the last decades and the great interest to equations with discontinuous coefficients. The present dissertation is dedicated to the study of two kinds of parabolic equations with discontinuous data. There are considered various boundary problems for nonlinear operators satisfying the Campanato condition (nonlinear variant of the Cordes condition) and there are proved strong solvability in the Sobolev spaces. A special attention is paid to the oblique derivative problem for linear and quasilinear uniformly parabolic operators with VMO coefficients. There are proved existence and uniqueness results in the Sobolev and Morrey spaces and H¨older continuity of the solution and its spatial gradient.
Parabolic Equations with Discontinuous Coefficients, Ph.D. Thesis
SOFTOVA Lyoubomira
2001
Abstract
The partial differential equations are a fundamental tool in the contemporary applied mathematics. Their importance is explained by the fact that various physical, chemical and other real processes are mathematically modeled through differential equations. On the other hand, the jump character of a grate part of the natural phenomena logically leads to their describing by problems with discontinuous data. This explains the rapid growth of the theory of PDE in the last decades and the great interest to equations with discontinuous coefficients. The present dissertation is dedicated to the study of two kinds of parabolic equations with discontinuous data. There are considered various boundary problems for nonlinear operators satisfying the Campanato condition (nonlinear variant of the Cordes condition) and there are proved strong solvability in the Sobolev spaces. A special attention is paid to the oblique derivative problem for linear and quasilinear uniformly parabolic operators with VMO coefficients. There are proved existence and uniqueness results in the Sobolev and Morrey spaces and H¨older continuity of the solution and its spatial gradient.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
