Free Boundary Problems, 1st ed. 2018
Regularity Properties Near the Fixed Boundary
Lecture Notes in Mathematics Series, Vol. 2218

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Language: English

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This book is concerned with several elliptic and parabolic obstacle-type problems with a focus on the cases where the free and fixed boundaries meet. The results presented complement those found in existing books in the subject, which mainly treat regularity properties away from the fixed boundary.

The topics include optimal regularity, analysis of global solutions, tangential touch of the free and fixed boundaries, as well as Lipschitz- and $C^1$-regularity of the free boundary. Special attention is given to local versions of various monotonicity formulas.

The intended audience includes research mathematicians and advanced graduate students interested in problems with free boundaries.
- Introduction. - No-Sign Parabolic Obstacle-Type Problems. - Boundary Estimates for Solutions of Free Boundary Problems.
Darya Apushkinskaya received her PhD degree from St. Petersburg State University and her Habilitation from Saarland University. Her research includes problems in non-smooth domains for equations with singular coefficients, variational problems with non-standard growth conditions, error analysis for solutions of variational and nonvariational problems and, of course,  free boundary problems and their applications to biology and solid state physics.

Presents obstacle-type problems in the parabolic case, not just the elliptic case

Focuses on the cases where the free and fixed boundaries meet

Several appendices provide the reader with a detailed exposition of the deep techniques of the theory