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A Geometric Approach to Free Boundary Problems, PDF eBook

A Geometric Approach to Free Boundary Problems PDF

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Free boundary (or moving boundary or phase transition) problems surface in many areas of analysis, geometry, and applied mathematics.

A typical example is the evolving interphase between a solid and liquid phase: if we know the initial configuration well enough, we should be able to reconstruct its evolution, in particular, the evolution of the interphase.

In this book we present a series of ideas, methods, and techniques for treating the most basic issues of such a problem.

In particular, we describe the very fundamental tools of geometry and real analysis that make this possible: properties of harmonic and caloric measures in Lipschitz domains, a relation between parallel surfaces and elliptic equations, monotonicity formulas and rigidity, etc.

We hope that the tools and ideas presented here will serve as a basis for the study of more complex phenomena and problems.

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