The Method of Layer Potentials for the Heat Equation in Time-Varying Domains
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Recent years have seen renewed interest in the solution of parabolic boundary value problems by the method of layer potentials, a method that has been extraordinarily useful in the solution of elliptic problems. This book develops this method for the heat equation in time-varying domains. In the first chapter, Lewis and Murray show that certain singular integral operators on L] p are bounded. In the second chapter, they develop a modification of the David buildup scheme, as well as some extension theorems, to obtain L]p boundedness of the double layer heat potential on the boundary of the doma...
Recent years have seen renewed interest in the solution of parabolic boundary value problems by the method of layer potentials, a method that has been extraordinarily useful in the solution of elliptic problems. This book develops this method for the heat equation in time-varying domains. In the first chapter, Lewis and Murray show that certain singular integral operators on L] p are bounded. In the second chapter, they develop a modification of the David buildup scheme, as well as some extension theorems, to obtain L]p boundedness of the double layer heat potential on the boundary of the domains. The third chapter uses the results of the first two, along with a buildup scheme, to show the mutual absolute continuity of parabolic measure and a certain projective Lebesgue measure. Lewis and Murray also obtain A infinity results and discuss the Dirichlet and Neumann problems for a certain subclass of the domains.