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Journal ArticleDOI

The one phase free boundary problem for the p-Laplacian with non-constant Bernoulli boundary condition

14 Feb 2002-Transactions of the American Mathematical Society (American Mathematical Society (AMS))-Vol. 354, Iss: 6, pp 2399-2416
TL;DR: In this paper, it was shown that the exterior and interior free boundary problem with a Bernoulli law, with a prescribed pressure a(x) on the free streamline of the flow, has convex solutions provided the initial domains are convex.
Abstract: Our objective, here, is to generalize our earlier results on the existence of classical convex solution to a free boundary problem with a Bernoulli-type boundary gradient condition and with the p-Laplacian as the governing operator. The main theorems of this paper assert that the exterior and the interior free boundary problem with a Bernoulli law, i.e. with a prescribed pressure a(x) on the free streamline of the flow, have convex solutions provided the initial domains are convex. The continuous function a(x) is subject to certain convexity properties. In our earlier results we have considered the case of constant a(x). In the lines of the proof of the main results we also prove the semi-continuity (up to the boundary) of the gradient of the p-capacitary potentials in convex rings, with C-1 boundaries.

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Citations
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Journal ArticleDOI
TL;DR: In this article, a uniform Lipschitz regularity of uniformly bounded solutions for the singular perturbation problem was shown. But the uniform Lipinski regularity was not shown for the stationary case of a combus- tion problem with a nonlinearity of power type.
Abstract: In this paper we initiate the study of the nonlinear one phase singular perturbation problem divNjru " j p 2 ru " OE"Nu " O; N1

35 citations

Journal ArticleDOI
TL;DR: The numerical solution of the free boundary Bernoulli problem is addressed and an iterative method based on a level-set formulation and boundary element method is proposed.
Abstract: The numerical solution of the free boundary Bernoulli problem is addressed. An iterative method based on a level-set formulation and boundary element method is proposed. Issues related to the implementation, the accuracy, and the generality of the method are discussed. The efficiency of the approach is illustrated by numerical results.

29 citations


Cites background from "The one phase free boundary problem..."

  • ...1The result is in fact established in the more general case of the p-Laplacian in [18] and is generalized to non-constant μ in [19]....

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Journal ArticleDOI
TL;DR: In this paper, the authors studied overdetermined boundary conditions for positive solutions to the p Laplacian in a bounded domain D. They showed that these conditions imply uniqueness in certain free boundary problems.
Abstract: We study overdetermined boundary conditions for positive solutions to the p Laplacian in a bounded domain D. We show these conditions imply uniqueness in certain free boundary problems.

22 citations


Additional excerpts

  • ...…u defined on a domain D = D a p ⊃ F with ∗ · u p−2 u = 0 weakly in D\F ∗∗ u x → 1 whenever x → y ∈ F and u x → 0 as x → y ∈ D (1.6) ∗ ∗ ∗ u x → a whenever x → y ∈ D This problem was solved in Henrot and Shahgholian (2000a) (see also Henrot and Shahgholian, 2000b, 2002 for related problems)....

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Journal ArticleDOI
TL;DR: In this paper, the authors prove concavity properties connected to nonlinear Bernoulli type free boundary problems and study the behavior of the free boundary with respect to the given boundary data.
Abstract: We prove some concavity properties connected to nonlinear Bernoulli type free boundary problems. In particular, we prove a Brunn–Minkowski inequality and an Urysohn’s type inequality for the Bernoulli Constant and we study the behavior of the free boundary with respect to the given boundary data. Moreover we prove a uniqueness result regarding the interior problem.

21 citations


Cites background from "The one phase free boundary problem..."

  • ...The treatment of the nonlinear case is more recent and mainly due to Henrot and Shahgholian, see for instance [15]-[18]; see also [3], [14], [23] and references therein....

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Journal ArticleDOI
TL;DR: In this paper, the authors studied overdetermined boundary conditions for positive solutions to some elliptic partial differential equations of the -Laplacian type in a bounded domain and showed that these conditions imply uniform rectifiability of the solution and also yield the solution to certain symmetry problems.
Abstract: We study overdetermined boundary conditions for positive solutions to some elliptic partial differential equations of -Laplacian type in a bounded domain . We show that these conditions imply uniform rectifiability of and also that they yield the solution to certain symmetry problems.

19 citations


Cites background from "The one phase free boundary problem..."

  • ...This problem was solved in [10] (see also [11, 12] for related problems)....

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References
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Book
07 Jan 2013
TL;DR: In this article, Leray-Schauder and Harnack this article considered the Dirichlet Problem for Poisson's Equation and showed that it is a special case of Divergence Form Operators.
Abstract: Chapter 1. Introduction Part I: Linear Equations Chapter 2. Laplace's Equation 2.1 The Mean Value Inequalities 2.2 Maximum and Minimum Principle 2.3 The Harnack Inequality 2.4 Green's Representation 2.5 The Poisson Integral 2.6 Convergence Theorems 2.7 Interior Estimates of Derivatives 2.8 The Dirichlet Problem the Method of Subharmonic Functions 2.9 Capacity Problems Chapter 3. The Classical Maximum Principle 3.1 The Weak Maximum Principle 3.2 The Strong Maximum Principle 3.3 Apriori Bounds 3.4 Gradient Estimates for Poisson's Equation 3.5 A Harnack Inequality 3.6 Operators in Divergence Form Notes Problems Chapter 4. Poisson's Equation and Newtonian Potential 4.1 Holder Continuity 4.2 The Dirichlet Problem for Poisson's Equation 4.3 Holder Estimates for the Second Derivatives 4.4 Estimates at the Boundary 4.5 Holder Estimates for the First Derivatives Notes Problems Chapter 5. Banach and Hilbert Spaces 5.1 The Contraction Mapping 5.2 The Method of Cintinuity 5.3 The Fredholm Alternative 5.4 Dual Spaces and Adjoints 5.5 Hilbert Spaces 5.6 The Projection Theorem 5.7 The Riesz Representation Theorem 5.8 The Lax-Milgram Theorem 5.9 The Fredholm Alternative in Hilbert Spaces 5.10 Weak Compactness Notes Problems Chapter 6. Classical Solutions the Schauder Approach 6.1 The Schauder Interior Estimates 6.2 Boundary and Global Estimates 6.3 The Dirichlet Problem 6.4 Interior and Boundary Regularity 6.5 An Alternative Approach 6.6 Non-Uniformly Elliptic Equations 6.7 Other Boundary Conditions the Obliue Derivative Problem 6.8 Appendix 1: Interpolation Inequalities 6.9 Appendix 2: Extension Lemmas Notes Problems Chapter 7. Sobolev Spaces 7.1 L^p spaces 7.2 Regularization and Approximation by Smooth Functions 7.3 Weak Derivatives 7.4 The Chain Rule 7.5 The W^(k,p) Spaces 7.6 DensityTheorems 7.7 Imbedding Theorems 7.8 Potential Estimates and Imbedding Theorems 7.9 The Morrey and John-Nirenberg Estimes 7.10 Compactness Results 7.11 Difference Quotients 7.12 Extension and Interpolation Notes Problems Chapter 8 Generalized Solutions and Regularity 8.1 The Weak Maximum Principle 8.2 Solvability of the Dirichlet Problem 8.3 Diferentiability of Weak Solutions 8.4 Global Regularity 8.5 Global Boundedness of Weak Solutions 8.6 Local Properties of Weak Solutions 8.7 The Strong Maximum Principle 8.8 The Harnack Inequality 8.9 Holder Continuity 8.10 Local Estimates at the Boundary 8.11 Holder Estimates for the First Derivatives 8.12 The Eigenvalue Problem Notes Problems Chapter 9. Strong Solutions 9.1 Maximum Princiles for Strong Solutions 9.2 L^p Estimates: Preliminary Analysis 9.3 The Marcinkiewicz Interpolation Theorem 9.4 The Calderon-Zygmund Inequality 9.5 L^p Estimates 9.6 The Dirichlet Problem 9.7 A Local Maximum Principle 9.8 Holder and Harnack Estimates 9.9 Local Estimates at the Boundary Notes Problems Part II: Quasilinear Equations Chapter 10. Maximum and Comparison Principles 10.1 The Comparison Principle 10.2 Maximum Principles 10.3 A Counterexample 10.4 Comparison Principles for Divergence Form Operators 10.5 Maximum Principles for Divergence Form Operators Notes Problems Chapter 11. Topological Fixed Point Theorems and Their Application 11.1 The Schauder Fixes Point Theorem 11.2 The Leray-Schauder Theorem: a Special Case 11.3 An Application 11.4 The Leray-Schauder Fixed Point Theorem 11.5 Variational Problems Notes Chapter 12. Equations in Two Variables 12.1 Quasiconformal Mappings 12.2 holder Gradient Estimates for Linear Equations 12.3 The Dirichlet Problem for Uniformly Elliptic Equations 12.4 Non-Uniformly Elliptic Equations Notes Problems Chapter 13. Holder Estimates for

18,443 citations

Book ChapterDOI
01 Jan 1997
TL;DR: In this paper, a class of partial differential equations that generalize and are represented by Laplace's equation was studied. And the authors used the notation D i u, D ij u for partial derivatives with respect to x i and x i, x j and the summation convention on repeated indices.
Abstract: We study in this chapter a class of partial differential equations that generalize and are to a large extent represented by Laplace’s equation. These are the elliptic partial differential equations of second order. A linear partial differential operator L defined by $$ Lu{\text{: = }}{a_{ij}}\left( x \right){D_{ij}}u + {b_i}\left( x \right){D_i}u + c\left( x \right)u $$ is elliptic on Ω ⊂ ℝ n if the symmetric matrix [a ij ] is positive definite for each x ∈ Ω. We have used the notation D i u, D ij u for partial derivatives with respect to x i and x i , x j and the summation convention on repeated indices is used. A nonlinear operator Q, $$ Q\left( u \right): = {a_{ij}}\left( {x,u,Du} \right){D_{ij}}u + b\left( {x,u,Du} \right) $$ [D u = (D 1 u, ..., D n u)], is elliptic on a subset of ℝ n × ℝ × ℝ n ] if [a ij (x, u, p)] is positive definite for all (x, u, p) in this set. Operators of this form are called quasilinear. In all of our examples the domain of the coefficients of the operator Q will be Ω × ℝ × ℝ n for Ω a domain in ℝ n . The function u will be in C 2(Ω) unless explicitly stated otherwise.

8,299 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

Book
01 Jan 1972
TL;DR: In this paper, the authors define the notion of potentials and their basic properties, including the capacity and capacity of a compact set, the properties of a set of irregular points, and the stability of the Dirichlet problem.
Abstract: 1. Spaces of measures and signed measures. Operations on measures and signed measures (No. 1-5).- 2. Space of distributions. Operations on distributions (No. 6-10)..- 3. The Fourier transform of distributions (No. 11-13).- I. Potentials and their basic properties.- 1. M. Riesz kernels (No. 1-3).- 2. Superharmonic functions (No. 4-5).- 3. Definition of potentials and their simplest properties (No. 6-9)...- 4. Energy. Potentials with finite energy (No. 10-15).- 5. Representation of superharmonic functions by potentials (No. 16-18).- 6. Superharmonic functions of fractional order (No. 19-25).- II. Capacity and equilibrium measure.- 1. Equilibrium measure and capacity of a compact set (No. 1-5).- 2. Inner and outer capacities and equilibrium measures. Capacitability (No. 6-10).- 3. Metric properties of capacity (No. 11-14).- 4. Logarithmic capacity (No. 15-18).- III. Sets of capacity zero. Sequences and bounds for potentials.- 1. Polar sets (No. 1-2).- 2. Continuity properties of potentials (No. 3-4).- 3. Sequences of potentials of measures (No. 5-8).- 4. Metric criteria for sets of capacity zero and bounds for potentials (No. 9-11).- IV. Balayage, Green functions, and the Dirichlet problem.- 1. Classical balayage out of a region (No. 1-6).- 2. Balayage for arbitrary compact sets (No. 7-11).- 3. The generalized Dirichlet problem (No. 12-14).- 4. The operator approach to the Dirichlet problem and the balayage problem (No. 15-18).- 5. Balayage for M. Riesz kernels (No. 19-23)...- 6. Balayage onto Borel sets (No. 24-25).- V. Irregular points.- 1. Irregular points of Borel sets. Criteria for irregularity (No. 1-6)...- 2. The characteristics and types of irregular points (No. 7-8)...- 3. The fine topology (No. 9-11).- 4. Properties of set of irregular points (No. 12-15).- 5. Stability of the Dirichlet problem. Approximation of continuous functions by harmonic functions (No. 16-22).- VI. Generalizations.- 1. Distributions with finite energy and their potentials (No. 1-5)...- 2. Kernels of more general type (No. 6-11).- 3. Dirichlet spaces (No. 12-15).- Comments and bibliographic references.

1,885 citations


"The one phase free boundary problem..." refers methods in this paper

  • ...Since alsojrujjp is a subsolution to the operator Luj (Lemma 2.5) we can apply the comparison principle to obtainjrujjp vj in Sj .A sj !1we can invoke classical results on stability [ Lan ] (cf....

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Book
23 Jul 1993
TL;DR: In this article, a monograph evolved out of the 1990 Lipschitz Lectures presented by the author at the University of Bonn, Germany, recounts recent developments in the attempt to understand the local structure of the solutions of degenerate and singular parabolic partial differential equations.
Abstract: This monograph evolved out of the 1990 Lipschitz Lectures presented by the author at the University of Bonn, Germany. It recounts recent developments in the attempt to understand the local structure of the solutions of degenerate and singular parabolic partial differential equations.

1,694 citations


Additional excerpts

  • ...also [Di] for the parabolic case....

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