Free Boundary Problems of Obstacle Type, a Numerical and Theoretical Study
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Elliptic Pdes
viii CHAPTER 4 Elliptic PDEs One of the main advantages of extending the class of solutions of a PDE from classical solutions with continuous derivatives to weak solutions with weak deriva- tives is that it is easier to prove the existence of weak solutions. Having estab- lished the existence of weak solutions, one may then study their properties, such as uniqueness and regularity, and perhaps prove under appropriate assumptions that the weak solutions are, in fact, classical solutions. There is often considerable freedom in how one defines a weak solution of a PDE; for example, the function space to which a solution is required to belong is not given a priori by the PDE itself. Typically, we look for a weak formulation that reduces to the classical formulation under appropriate smoothness assumptions and which is amenable to a mathematical analysis; the notion of solution and the spaces to which solutions belong are dictated by the available estimates and analysis. 4.1. Weak formulation of the Dirichlet problem Let us consider the Dirichlet problem for the Laplacian with homogeneous boundary conditions on a bounded domain Ω in Rn, (4.1) −∆u = f in Ω, (4.2) u =0 on ∂Ω. First, suppose that the boundary of Ω is smooth and u,f : Ω → R are smooth functions. Multiplying (4.1) by a test function φ, integrating the result over Ω, and using the divergence theorem, we get ∞ (4.3) Du · Dφ dx = fφ dx for all φ ∈ Cc (Ω). ZΩ ZΩ The boundary terms vanish because φ = 0 on the boundary. -
Elliptic Pdes
CHAPTER 4 Elliptic PDEs One of the main advantages of extending the class of solutions of a PDE from classical solutions with continuous derivatives to weak solutions with weak deriva- tives is that it is easier to prove the existence of weak solutions. Having estab- lished the existence of weak solutions, one may then study their properties, such as uniqueness and regularity, and perhaps prove under appropriate assumptions that the weak solutions are, in fact, classical solutions. There is often considerable freedom in how one defines a weak solution of a PDE; for example, the function space to which a solution is required to belong is not given a priori by the PDE itself. Typically, we look for a weak formulation that reduces to the classical formulation under appropriate smoothness assumptions and which is amenable to a mathematical analysis; the notion of solution and the spaces to which solutions belong are dictated by the available estimates and analysis. 4.1. Weak formulation of the Dirichlet problem Let us consider the Dirichlet problem for the Laplacian with homogeneous boundary conditions on a bounded domain Ω in Rn, (4.1) −∆u = f in Ω; (4.2) u = 0 on @Ω: First, suppose that the boundary of Ω is smooth and u; f : Ω ! R are smooth functions. Multiplying (4.1) by a test function φ, integrating the result over Ω, and using the divergence theorem, we get Z Z 1 (4.3) Du · Dφ dx = fφ dx for all φ 2 Cc (Ω): Ω Ω The boundary terms vanish because φ = 0 on the boundary. -
An Obstacle Problem in a Plane Domain with Two Solutions
An obstacle problem in a plane domain with two solutions Giovany M. Figueiredo Universidade Federal do Par´a- UFPA Faculdade de Matem´atica CEP 66075-110, Bel´em, PA, Brasil E-mail: [email protected] Marcelo Furtado Universidade de Bras´ılia - UNB Departamento de Matem´atica CEP 70910-900, Bras´ıla, DF, Brasil E-mail: [email protected] and Marcelo Montenegro Universidade Estadual de Campinas, IMECC, Departamento de Matem´atica, Rua S´ergio Buarque de Holanda, 651 CEP 13083-859, Campinas, SP, Brasil E-mail: [email protected] Abstract We prove the existence of two nontrivial solutions for the variational inequality RΩ ∇u∇(v − u) ≥ RΩ f(u)(v − u) for every v belonging to some convex set, where Ω ⊂ R2. The function f has critical exponential growth, 2 in the sense that it behaves like exp(α0s ) as |s|→∞, for some α0 > 0. We use variational methods for lower semicontinuous functionals. Mathematics Subject Classification: 35J60, 35J65, 35J85, 49J40, 35R35, 47J20. Keywords: obstacle problem, exponential critical growth, Trudinger-Moser inequality. 1 Introduction The obstacle problem arises in many branches as for example in the elasticity theory, in control games, minimal surfaces, constrained heating and financial mathematics [6]. In a simple version, one is lead to minimize the functional (u)= u 2dx J |∇ | ZΩ 1 in the set = u H1(Ω) : u = h on ∂Ω and u ζ a.e. in Ω , K { ∈ ≥ } where Ω RN is an open bounded domain with smooth boundary, h is a smooth function⊂ defined on ∂Ω and the obstacle ζ is a smooth function defined in Ω such that ζ ∂Ω < h. -
Regularity for Obstacle Problems in Infinite Dimensional Hilbert
Advances in Mathematics 220 (2009) 964–983 www.elsevier.com/locate/aim Regularity for obstacle problems in infinite dimensional Hilbert spaces Andrzej Swi¸´ ech a,1, Eduardo V. Teixeira b,∗,2 a School of Mathematics, Georgia Institute of Technology, Atlantla, GA 30332, USA b Universidade Federal do Ceará, Departamento de Matemática, Fortaleza, CE, Brazil Received 9 March 2007; accepted 20 October 2008 Available online 13 November 2008 Communicated by L. Caffarelli Abstract In this paper we study fully nonlinear obstacle-type problems in Hilbert spaces. We introduce the notion of Q-elliptic equation and prove existence, uniqueness, and regularity of viscosity solutions of Q-elliptic obstacle problems. In particular we show that solutions of concave problems with semiconvex obstacles are 2,∞ in the space WQ . © 2008 Elsevier Inc. All rights reserved. MSC: primary 49L25, 35R15, 35R35 Keywords: Obstacle-type problems; Optimal regularity; Fully nonlinear equations in Hilbert spaces; Viscosity solutions 1. Introduction Given an elastic membrane u attached to a fixed boundary ∂D, a classical problem in math- ematical physics asks for the equilibrium position of the membrane when it is constrained to lie above a given obstacle ϕ. Mathematically this problem can be formulated as follows. Given a domain D, a function g : ∂D → R (the deformation of the fixed boundary) and ϕ : D → R * Corresponding author. E-mail addresses: [email protected] (A. Swi¸´ ech), [email protected] (E.V. Teixeira). 1 Supported by NSF grant DMS 0500270. 2 Has been partially supported by NSF grant DMS 0600930 and CNPq. 0001-8708/$ – see front matter © 2008 Elsevier Inc. -
Advanced Partial Differential Equations Prof. Dr. Thomas
Advanced Partial Differential Equations Prof. Dr. Thomas Sørensen summer term 2015 Marcel Schaub July 2, 2015 1 Contents 0 Recall PDE 1 & Motivation 3 0.1 Recall PDE 1 . .3 1 Weak derivatives and Sobolev spaces 7 1.1 Sobolev spaces . .8 1.2 Approximation by smooth functions . 11 1.3 Extension of Sobolev functions . 13 1.4 Traces . 15 1.5 Sobolev inequalities . 17 2 Linear 2nd order elliptic PDE 25 2.1 Linear 2nd order elliptic partial differential operators . 25 2.2 Weak solutions . 26 2.3 Existence via Lax-Milgram . 28 2.4 Inhomogeneous bounday value problems . 35 2.5 The space H−1(U) ................................ 36 2.6 Regularity of weak solutions . 39 A Tutorials 58 A.1 Tutorial 1: Review of Integration . 58 A.2 Tutorial 2 . 59 A.3 Tutorial 3: Norms . 61 A.4 Tutorial 4 . 62 A.5 Tutorial 6 (Sheet 7) . 65 A.6 Tutorial 7 . 65 A.7 Tutorial 9 . 67 A.8 Tutorium 11 . 67 B Solutions of the problem sheets 70 B.1 Solution to Sheet 1 . 70 B.2 Solution to Sheet 2 . 71 B.3 Solution to Problem Sheet 3 . 73 B.4 Solution to Problem Sheet 4 . 76 B.5 Solution to Exercise Sheet 5 . 77 B.6 Solution to Exercise Sheet 7 . 81 B.7 Solution to problem sheet 8 . 84 B.8 Solution to Exercise Sheet 9 . 87 2 0 Recall PDE 1 & Motivation 0.1 Recall PDE 1 We mainly studied linear 2nd order equations – specifically, elliptic, parabolic and hyper- bolic equations. Concretely: • The Laplace equation ∆u = 0 (elliptic) • The Poisson equation −∆u = f (elliptic) • The Heat equation ut − ∆u = 0, ut − ∆u = f (parabolic) • The Wave equation utt − ∆u = 0, utt − ∆u = f (hyperbolic) We studied (“main motivation; goal”) well-posedness (à la Hadamard) 1. -
Arxiv:1707.00992V1 [Math.AP] 4 Jul 2017
OBSTACLE PROBLEMS AND FREE BOUNDARIES: AN OVERVIEW XAVIER ROS-OTON Abstract. Free boundary problems are those described by PDEs that exhibit a priori unknown (free) interfaces or boundaries. These problems appear in Physics, Probability, Biology, Finance, or Industry, and the study of solutions and free boundaries uses methods from PDEs, Calculus of Variations, Geometric Measure Theory, and Harmonic Analysis. The most important mathematical challenge in this context is to understand the structure and regularity of free boundaries. In this paper we provide an invitation to this area of research by presenting, in a completely non-technical manner, some classical results as well as some recent results of the author. 1. Introduction Many problems in Physics, Industry, Finance, Biology, and other areas can be described by PDEs that exhibit apriori unknown (free) interfaces or boundaries. These are called Free Boundary Problems. A classical example is the Stefan problem, which dates back to the 19th century [54, 35]. It describes the melting of a block of ice submerged in liquid water. In the simplest case (the one-phase problem), there is a region where the temperature is positive (liquid water) and a region where the temperature is zero (the ice). In the former region the temperature function θ(t; x) solves the heat equation (i.e., θt = ∆θ in f(t; x): θ(t; x) > 0g), while in the other region the temperature θ is just zero. The position of the free boundary that separates the two regions is part of the problem, and is determined by an extra boundary condition on such interface 2 (namely, jrxθj = θt on @fθ > 0g). -
Free Boundary Regularity in Obstacle Problems
Free boundary regularity in obstacle problems Alessio Figalli Abstract. These notes record and expand the lectures for the \Journ´ees Equations´ aux D´eriv´eesPartielles 2018" held by the author during the week of June 11-15, 2018. The aim is to give a overview of the classical theory for the obstacle problem, and then present some recent developments on the regularity of the free boundary. Contents 1. Introduction: the obstacle problem 1 2. The functional setting: existence and uniqueness 2 3. Euler-Lagrange equation and consequences 3 4. Optimal regularity 5 5. A new formulation of the obstacle problem 8 6. Non-degeneracy of solutions 9 7. Blow-up analysis and Caffarelli’s dichotomy 10 7.1. Regular free boundary points 11 7.2. Singular free boundary points 11 7.3. Caffarelli’s dichotomy theorem 13 8. Uniqueness of blow-up at singular points 16 9. Stratification and C1 regularity of the singular set 19 10. Recent developments 20 11. Future directions 24 Bibliography 25 1. Introduction: the obstacle problem The classical obstacle problem aims to describe the shape of an elastic membrane lying above an obstacle. n Mathematically, this problem can be described as follows: given a domain Ω ⊂ R , a function ' :Ω ! R (the obstacle), and a function f : @Ω ! R satisfying f ≥ 'j@Ω (the boundary condition), one wants to minimize the Dirichlet integral among all functions that coincide with f on @Ω and lie above the obstacle ' (see Figure 1). Hence, one is lead to minimize Z jrvj2 min : vj@Ω = f; v ≥ ' ; (1.1) v Ω 2 R jrvj2 where the Dirichlet integral Ω 2 represents the elastic energy of the membrane corre- sponding to the graph of v. -
Delta Functions and Distributions
When functions have no value(s): Delta functions and distributions Steven G. Johnson, MIT course 18.303 notes Created October 2010, updated March 8, 2017. Abstract x = 0. That is, one would like the function δ(x) = 0 for all x 6= 0, but with R δ(x)dx = 1 for any in- These notes give a brief introduction to the mo- tegration region that includes x = 0; this concept tivations, concepts, and properties of distributions, is called a “Dirac delta function” or simply a “delta which generalize the notion of functions f(x) to al- function.” δ(x) is usually the simplest right-hand- low derivatives of discontinuities, “delta” functions, side for which to solve differential equations, yielding and other nice things. This generalization is in- a Green’s function. It is also the simplest way to creasingly important the more you work with linear consider physical effects that are concentrated within PDEs, as we do in 18.303. For example, Green’s func- very small volumes or times, for which you don’t ac- tions are extremely cumbersome if one does not al- tually want to worry about the microscopic details low delta functions. Moreover, solving PDEs with in this volume—for example, think of the concepts of functions that are not classically differentiable is of a “point charge,” a “point mass,” a force plucking a great practical importance (e.g. a plucked string with string at “one point,” a “kick” that “suddenly” imparts a triangle shape is not twice differentiable, making some momentum to an object, and so on. -
Arxiv:1905.06146V5 [Math.AP] 5 Jun 2020
THE OBSTACLE PROBLEM FOR A CLASS OF DEGENERATE FULLY NONLINEAR OPERATORS JOAO˜ VITOR DA SILVA AND HERNAN´ VIVAS Abstract. We study the obstacle problem for fully nonlinear elliptic operators with an anisotropic degeneracy on the gradient: min f − |Du|γ F (D2u), u − φ = 0 in Ω u = g on ∂Ω for some degeneracy parameter γ ≥ 0, uniformly elliptic operator F , bounded source term f, and suitably smooth obstacle φ and boundary datum g. We obtain exis- tence/uniqueness of solutions and prove sharp regularity estimates at the free bound- ary points, namely ∂{u>φ} ∩ Ω. In particular, for the homogeneous case (f ≡ 0) we get that solutions are C1,1 at free boundary points, in the sense that they detach from the obstacle in a quadratic fashion, thus beating the optimal regularity allowed for such degenerate operators. We also prove several non-degeneracy properties of solutions and partial results regarding the free boundary. These are the first results for obstacle problems driven by degenerate type operators in non-divergence form and they are a novelty even for the simpler prototype given by an operator of the form G[u]= |Du|γ ∆u, with γ > 0 and f ≡ 1. 1. Introduction 1.1. Motivation and main proposals. A classical problem from Mathematical Physics refers to the equilibrium position of an elastic membrane (whose boundary is held fixed) lying on top of a given body (an obstacle) under the action of contact and/or action-at-a- distance forces, e.g. friction, tension, air resistance and gravity. Currently, this archetype model is often called an obstacle problem. -
Generalized Solutions, Sobolev Spaces (2017)
AMATH 731: Applied Functional Analysis Fall 2017 Generalized solutions, Sobolev spaces (To accompany Section 4.6 of the AMATH 731 Course Notes) Introduction As a simple motivating example, consider the following inhomogeneous DE of the form Lu = f, (Lu)(x)= u′′(x)+ g(x)u(x)= f(x). (1) In the usual introductory courses of ODEs, f(x) is assumed sufficiently “nice,” i.e., at least continuous, so that u′′(x) is continuous, implying that u(x) is twice differentiable. At worst, f(x) could be piecewise continuous, which still does not represent a problem. One may then resort to a number of “classical” techniques to solve for u(x), or at least approximations to it (e.g., Fourier series, power series). But what if f(x) is not so “nice”? What if f(x) is discontinuous or, at best, integrable? Classical methods will break down here. One could, however, still try to use Fourier series (assuming that the Fourier expansion of f(x) exists). But the question remains: In what space does the solution u(x) live? What does it mean to say that u′(x) or u′′(x) is a function in L2? This is a fundamental problem in the study of partial differential equations (PDEs). Only a small class of partial differential equations, e.g., Laplace’s equation, can be solved in the classical sense. Many others, if not most, cannot. One must resort to other methods to construct “less smooth” solutions. An example of such a method is that of generalized or weak solutions. Sobolev spaces represent the completion of the appropriate classical function spaces in order to accomodate such solutions. -
Weak Solutions of Nonlinear Evolution Equations of Sobolev-Galpern Type
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OFDIFFERENTIAL EQUATIONS 11, 252-265 (1972) Weak Solutions of Nonlinear Evolution Equations of Sobolev-Galpern Type R. E. SHOWALTER Department of Mathematics, University of Texas, Austin, Texas 78712 Received April 30, 1970 1. INTRODUCTION Consider the abstract evolution equation which may arise from a partial differential equation of order 2m + 1 in which W(t)) and WN are families of linear elliptic operators of order 2m and F contains derivatives of order \cm. The objective here is to formulate a Cauchy problem for (1 .l) and to prove the existence and uniqueness of a solution by Hilbert space methods. Weak and strong solutions of (1.1) are specified in Section 2 where the uniqueness of a weak solution is established. A resolving operator is constructed in Section 3; this operator provides the weak solution of the homogeneous form of (1.1) and suggests the concept of a mild solution, essentially an integrated form of (1.1). The existence of a mild and weak solution of the nonlinear Cauchy problem is established in Section 4. No explicit requirements on the domains of the unbounded operators in (1 .l) are necessary, and the dependence of these operators on the time-parameter is essentially a measurability assumption. The abstract problem we consider here arises as a realization of various partial differential equations from problems in mathematical physics [2, 5, 14, 25, 26, 271. Following [17], we say that a partial differential equation is of Sobolev-Galpern type if the highest order terms contain derivatives in both space and time coordinates. -
Arxiv:1903.00429V1
ON GRADIENT FLOWS WITH OBSTACLES AND EULER’S ELASTICA MARIUS MULLER¨ Abstract. We examine a steepest energy descent flow with obstacle con- straint in higher order energy frameworks where the maximum principle is not available. We construct the flow under general assumptions using De Giorgi’s minimizing movement scheme. Our main application will be the elastic flow of graph curves with Navier boundary conditions for which we study long-time existence and asymptotic behavior. 1. Introduction Under Obstacle Problems one understands the question whether a given energy functional attains a minimum in the set Cψ : u V : u x ψ x a.e. , where V is a space of Lebesgue-measurable functions“ and t ψP is a certainp q ě measurablep q u func- tion such that Cψ , the so-called obstacle. The question can be posed more generally as minimization‰ H of an energy functional in some convex closed subset of a Banach space V , but since Cψ has more structural properties than just convex- ity and closedness, one could also impose more conditions on C. Part of the goal of this article is to identify reasonable structural requirements in a more abstract framework, e.g. if V is not required to be a space of functions. Such obstacle problems have various applications in physics and finance. A very important concept in this field is that of a variational inequality. This can be stated as follows: If V is a Banach Space, C V is convex and closed and E C1 V, R is an energy functional then Ă P p q u C is such that E u inf E w 0 DE u v u v C, w C P p q“ P p q ñ ď p qp ´ q @ P arXiv:1903.00429v1 [math.AP] 1 Mar 2019 where DE : V V 1 denotes the Frech´et derivative of E.