H. W. Alt and L. A. Caarelli. Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math., 325:105-144, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....may see, for instance, BZ] and [Se] for a classical introduction, and also [ACF1] and [ACF3] for recent developments. In particular, the first term in the functional (as shown, for instance, in Lemma 2. 4 below) takes into account the Equation of continuity, while the second term (as shown in [AC]) expresses Bernoulli s law on the free surface. The function u has the physical meaning of a stream function. The particular case a i,j = # i,j corresponds to a homogeneous medium. The functional in (1.1) was also dealt with in the theory of minimal surfaces: see, for instance, Md] and [CC] ....

....class A minimizers. We say that a minimizer is class A minimizer if the functional does not increase under any compact (but not necessarily small) modification (see Definition 7.5 below) The proof of Theorem 1. 1 will make use of the results of existence and rigidity of minimizers of [AC], the regularity result of [GG] several density estimates of [CC] and a geometric construction of [CL] In the second part of this paper, we will extend the results of Theorem 1.1 to more general Ginzburg Landau type functionals (see below Section 8) We remark that the results in the second ....

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Alt, H. W.; Ca#arelli, L. A. -- Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105--144.

....inviscid fluids. In this setting, the function u has the physical meaning of a stream function; the first term in the functional takes into account the Equation of continuity, while the second term expresses Bernoulli s law on the free surface: this is shown in Lemma 3.2. 2 here and in [AC]. The particular case a i,j = # i,j corresponds to a homogeneous medium. We will give a sketchy physical motivation in Section 1.3. For a more accurate reading, the interested reader may see, for instance, BZ] and [Se] for a classical introduction, and also [ACF1] and [ACF3] for recent ....

....v = 0, therefore the particles of fluid move along the level sets of u. Also, using the Equation of continuity and Bernoulli s law, one sees that #u = 0 (1.4) in the region occupied by the fluid jet, and that 2 p = 2 p = const (1.5) on the boundary of the jet. One can prove (see [AC] and Lemma 3.2.2 below) that the minimizers of the functional Q# ( 1,1) u) satisfy (1.4) and (1.5) above, for a suitable choice of Q. 1.3.2 Phase transition models Let us consider some kind of material that has two phases, that we label by 1 and 1. One may expect that the equilibria ....

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Alt, H. W.; Ca#arelli, L. A. -- Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105--144.

....modified nonlinear problem introduced in Section 3.1 has a solution, we use the following fixed point theorem: THEOREM 3. 6 (Theorem 11.3 of [7] Let T be a compact mapping of a Banach space B into itself, and suppose that there exists a constant M such that #u# B #M for all u # B and # # [0, 1] satisfying u = #Tu. Then T has a fixed point. We define T : H ( # 1 ) 1 # # H ( # 1 ) 1 # by letting u = Tw be the unique weak solution of the linear mixed boundary problem (3.8) that we have just constructed. By Theorem 3.2, T (H ( # 1 ) 1 # ) # H ( #) 1 # S , so the operator T is ....

....# S or # 1 until we consider the free boundary problem in the next section. A TRANSONIC SHOCK FREE BOUNDARY PROBLEM 17 We need to show that there exists an M 0 such that #u# B #M for all functions u # B # H ( # 1 ) 1 # S 2 that solve # Qu = 0 in W , # N u = # on S , # # [0, 1] , 3.19) u = u 1 on # , for the modified nonlinear operators of equation (3.5) Before proceeding with this estimate, we again replace u by u u 1 and solve the same mixed boundary value problem but with homogeneous Dirichlet data on # and with #(u) still centered at u 1 , so #(u) u 1 ....

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Alt, H. W.; Caffarelli, L. A. Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325 (1981), 105--144.

....Third author supported by the Swedish Natural Science Research Council. 1 2 JUAN MANFREDI, ARSHAK PETROSYAN, AND HENRIK SHAHGHOLIAN We refer to the paper of A. Acker and R. Meyer [AM] for background and further references. We also mention the pioneering work of H. W. Alt and L. A. Ca arelli [AC] in this connection. For p 2 (1; 1) under suitable convexity assumptions on a(x) and regularity assumptions on K to be speci ed later, problem (FB p ) admits a unique classical solution, which we denote(u p ; p ) see [HS3] The main purpose of this paper is to study the behavior of the pair ....

H. W. Alt and L. A. Caarelli,Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105-144.

....in the literature; in fact, even for p = 2, there is no existence proof for a classical solution valid in higher dimensions. Early existence results due to Beurling [B] Daniljuk [D] and Lavrent ev [LV] apply only for p = 2 and N = 2. The wellknown existence results of Alt and Caffarelli [AC] are applicable only for p = 2, and these solutions are not necessarily classical for N 3. An existence theorem for classical solutions in the starlike case for p = 2, a(x) constant, and N 2 was stated by Lacey and Shillor [LS] but their proof is not valid for N 3, because it is actually an ....

.... D g. Lemma 3.6 Problem 3.5 has a solution, U , which is Lipschitz continuous and has compact support on Omega = R N nCl(D ) and satisfies 0 U 1 in Omega and 4U = 0 in fU 0g: Furthermore, U satisfies the free boundary condition of Problem 1.1 in a certain weak sense. Proof. See [AC], Theorems 1.3 and 3.3, Lemmas 2.8, 2.3, and 2.4, and Theorem 2.5. 2 Lemma 3.7 If D is starlike with respect to all points in B ffi (0) then so is fU g for all 0 1; where U denotes a solution of Problem 3.5. Proof. For r 1, let a r (x) 1=r)a(x=r) U r (x) U(x=r) U r (x) ....

Alt, H. W. & Caffarelli, L. A., Existence and regularity for a minimum problem with a free boundary, J. Reine Angew. Math., 33(1984), 213-247.

....the notion of weak solutions of (6, 7) we investigate the stationary points of E (Definition 0.1 b) This will motivate the notion of weak solutions and highlight the difficulties we will encounter when proving their existence. The local minima of E have been studied in the multidimensional case [2]. The notion of stationary point depends on the set of admissible variations. To motivate Definition 0.1 b) we go back to the level of the Hele Shaw problem. For the Hele Shaw problem, which models an incompressible flow of a fluid, the set of admissible variations of a configuration Omega is ....

H.--W. Alt, L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angewandte Math. , 325 (1981), 105 -- 144.

....(0.3) Goodman [20] showed that 0.22# A 0.50#. The lower bound which he obtained was generated by a domain of the type shown in Figure 3. Figure 3 Later, Goodman and Reich [22] gave an improved upper bound of 0.38# for A. Lewis had shown in [31] by applying some deep results of Alt and Ca#arelli [2] in partial di#erential equations for free boundary problems that the extremal domains had to have piecewise analytic boundaries and that our methods developed in [10] for these omitted value problems could be used to give a geometric description for the boundaries of the extremal domains as ....

Alt, H., and Ca#arelli, L. "Existence and regularity for a minimum problem with free boundary. " J. Reine Angew. Math. 325 (1981) pp. 105-144.

....cost functional and to find a formulation which leads to optimally conditioned shape hessian at the solution. We also analyze and compare some fixed point type methods inspired by the shape optimization problems. 1. Introduction In this work we shall consider the famous Alt Caffarelli problem, [1]. That is, the problem of finding Sigma, the free boundary, so that 8 : Gamma Delta u = 0 in Omega ; u = 1 on Gamma; u = 0 on Sigma; u n = on Sigma (1.1) for given 0 and Gamma, see Figure 1. It is well known, 1] that the solution of (1.1) is a critical point of the ....

....we shall consider the famous Alt Caffarelli problem, 1] That is, the problem of finding Sigma, the free boundary, so that 8 : Gamma Delta u = 0 in Omega ; u = 1 on Gamma; u = 0 on Sigma; u n = on Sigma (1.1) for given 0 and Gamma, see Figure 1. It is well known, [1], that the solution of (1.1) is a critical point of the following energy E( Omega ; u) 1 2 Z Omega ruru 1 2 Z Omega 2 (1.2) with respect to Omega and u 2 V( Omega Gamma = Phi v 2 H 1( Omega Gamma j vj Gamma = 1; vj Sigma = 0 Psi . If is a negative constant and Gamma is a ....

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H.W. Alt, L.A. Cafarelli, Existence and regularity for a minimum problem with free boundary, J. Reine angew. Math 325 (1981), 105-144.

....connected Lipschitz domain, and u : Omega R is subjected to the volume constraints L N (fu = 0g) ff and L N (fu = 1g) fi: 1. 1) Here L N denotes the N dimensional Lebesgue measure in R N and ff; fi 0 satisfy ff fi L N( Omega Gamma2 Previous works by Alt and Caffarelli [3] and Aguilera, Alt and Caffarelli [2] address a similar problem where only one volume constraint is present and Dirichlet boundary conditions are imposed on u. They obtain existence of minimizers for I and regularity properties for solutions as well as their free boundaries. In our context, and in ....

Alt, H.W. and L.A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105--144.

.... (w) Gamma Z Qwds: Then the weak formulation of the free boundary problem (P) is the following variational problem: V ) Find (OE; u) OE is analytic, u 2 1 Phi H 1 D( Omega OE ) uj OE = 0 such that a OE (u; w) OE(w) for all w 2 H 1 D( Omega OE ) Remark. Following [3], the existence result of Problem (FF) or (P) can be established in a very weak sense, namely, u 2 H 1( Omega Gamma6 Thus, if we find a solution of problem (V ) we actually find the solution of problem (P) To finish this section we list some more notations. M = max x2I j (x) Gamma OE ....

H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., 325 (1981), pp. 105-144.

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H. W. Alt and L. A. Caarelli. Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math., 325:105-144, 1981.

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H.W. Alt, L.A. Cafarelli, Existence and regularity for a minimum problem with free boundary, J. Reine angew. Math 325 (1981), 105-144.

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