Free boundary problems. Regularity properties near the fixed boundary
This book deals with the obstacle-type free boundary problems. Several elliptic and parabolic one-phase and two-phase problems are studied. Particular attention is paid to the case when the contact point between free and fixed boundaries may arise. It is assumed that the Dirichlet boundary condition is set on the fixed boundary. The author establishes the optimal regularity of the solutions and, for the case of homogeneous Dirichlet condition, gives the classification of global solutions. Estimates of the higher derivatives of solutions in a neighborhood of the fixed boundary are obtained for various elliptic and parabolic obstacle-type problems. Careful analysis of the regularity properties of the free boundary in a neighborhood of the contact point is given. It is proved that there are no singular points on the free boundary near the fixed boundary. Lipschitz regularity and \(C^1\) regularity of the free boundary are investigated. To illustrate that some regularity properties may fail to take place at the contact point, the author gives the counterexample, in which the free boundary intersects the fixed boundary transversally in t-direction. One of the main technical tools in this book is monotonicity formulas, the author gives all the necessary information about various monotonicity formulas in the Appendix.
scientific article; zbMATH DE number 6927917