Topic
Obstacle problem
About: Obstacle problem is a research topic. Over the lifetime, 1792 publications have been published within this topic receiving 36903 citations.
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Book•
01 Jan 1980
TL;DR: In this paper, the SIAM edition Preface Glossary of notations Introduction Part I. Variational Inequalities in Rn Part II. Free Boundary Problems Governed by Elliptic Equations and Systems Part VII. A One Phase Stefan Problem Bibliography Index.
Abstract: Preface to the SIAM edition Preface Glossary of notations Introduction Part I. Variational Inequalities in Rn Part II. Variational Inequalities in Hilbert Space Part III. Variational Inequalities for Monotone Operators Part IV. Problems of Regularity Part V. Free Boundary Problems and the Coincidence Set of the Solution Part VI. Free Boundary Problems Governed by Elliptic Equations and Systems Part VII. Applications of Variational Inequalities Part VIII. A One Phase Stefan Problem Bibliography Index.
4,107 citations
01 Jan 1994
2,202 citations
Book•
16 May 2018
TL;DR: In this paper, the existence of solutions for the obstacle problem is investigated and the John-Nirenberg lemma is shown to be true for nonlinear potential theory with respect to a super-harmonic function.
Abstract: Introduction. 1: Weighted Sobolev spaces. 2: Capacity. 3: Supersolutions and the obstacle problem. 4: Refined Sobolev spaces. 5: Variational integrals. 6: A-harmonic functions. 7: A superharmonic functions. 8: Balayage. 9: Perron's method, barriers, and resolutivity. 10: Polar sets. 11: A-harmonic measure. 12: Fine topology. 13: Harmonic morphisms. 14: Quasiregular mappings. 15: Ap-weights and Jacobians of quasiconformal mappings. 16: Axiomatic nonlinear potential theory. Appendix I: The existence of solutions. Appendix II: The John-Nirenberg lemma. Bibliography. List of symbols. Index
2,017 citations
TL;DR: In this article, reflected solutions of one-dimensional backward stochastic differential equations are studied and the authors prove uniqueness and existence both by a fixed point argument and by approximation via penalization.
Abstract: We study reflected solutions of one-dimensional backward stochastic differential equations. The “reflection” keeps the solution above a given stochastic process. We prove uniqueness and existence both by a fixed point argument and by approximation via penalization. We show that when the coefficient has a special form, then the solution of our problem is the value function of a mixed optimal stopping–optimal stochastic control problem. We finally show that, when put in a Markovian framework, the solution of our reflected BSDE provides a probabilistic formula for the unique viscosity solution of an obstacle problem for a parabolic partial differential equation.
781 citations