In this paper I prove some conjectures of GURTIN [15] concerning the Van der Waals-Cahn-Hilliard theory of phase transitions. Consider a fluid, under isothermal conditions and confined to a bounded container 12 Q R', whose Gibbs free energy, per unit volume, is a prescribed function Wo of the density distribution u. The classical problem (cf. GURTIN [16]) of determining the stable configurations of the fluid is to minimize the total energy of the fluid, E(u) = f Wo(u(x)) dx, t2

/pdf/the-gradient-theory-of-phase-transitions-and-the-minimal-4hr0iaz1vl.pdf

The gradient theory of phase transitions and the minimal interface criterion

We study the nonlinear monotone elliptic problem  −div (a(x,∇u)) = μ on Ω, u = 0 on ∂Ω, when Ω ⊂ R , μ is a Radon measure with bounded total variation on Ω, 1 < p ≤ N , and u 7→ −div (a(x,∇u)) is a monotone operator acting on W 1,p 0 (Ω). We introduce a new definition of solution (the renormalized solution) in four equivalent ways. We prove the existence of a renormalized solution by an approximation procedure, where the key point is a stability result (the strong convergence in W 1,p 0 (Ω) of the truncates). We also prove partial uniqueness results. SISSA, via Beirut 4, 34014 Trieste, Italy Laboratoire d’Analyse Numerique, Universite Paris VI, Case 187, 75252 Paris cedex 05, France Dipartimento di Matematica, Universita di Roma I, P.le A. Moro 2, 00185 Roma, Italy Universite d’Orleans, rue de Chartres, B.P. 6759, 45067 Orleans cedex 2, France

/pdf/renormalized-solutions-of-elliptic-equations-with-general-2pld7ewfok.pdf

Renormalized solutions of elliptic equations with general measure data

We study the existence of positive solutions of the equation u" + a(t)f(u) = 0 with linear boundary conditions. We show the existence of at least one positive solution if f is either superlinear or sublinear by a simple application of a Fixed Point Theorem in cones.

/pdf/on-the-existence-of-positive-solutions-of-ordinary-4wta5reh16.pdf

On the existence of positive solutions of ordinary differential equations

Positive solutions of some nonlinear elliptic problems in exterior domains

Our main result is that, when $f$ is smooth and has bounded derivatives, and when $u$ belongs to the spaces $W^{s,p}$ and $W^{1,sp}$, the map $f(u)$ is in $W^{s,p}$.

https://hal.archives-ouvertes.fr/hal-00747694/document

Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces

/pdf/transformations-of-domains-in-the-plane-and-applications-in-42j8d3we7x.pdf

Transformations of domains in the plane and applications in the theory of functions

http://shelf2.library.cmu.edu/Tech/53220464.pdf

Existence and uniqueness results for semi-linear Dirichlet problems in annuli

Nemitsky operators on Sobolev spaces

/pdf/radial-averaging-transformations-with-various-metrics-23bebji02x.pdf

Radial averaging transformations with various metrics.

RADIAL AVERAGING OF DOMAINS, ESTIMATES FOR DIRICHLET INTEGRALS AND APPLICATIONS by Moshe Marcus Let & = {D,,...,D } be a family of domains in the plane, containing the origin. We define a radial averaging transformation R. on S by which we obtain a starlike domain D . When & is such that the domains D,,...^ are obtained from a fixed domain D by rotation or reflexion, ft. becomes j\ a radial symmetrization. One of the results we present is an inequality relating the conformal radius of D to the conformal radii of D, ,...,D at the origin. This result includes, as particular cases, the radial symmetrization results of Szego [11] (for starlike domains), Marcus [7] (for general domains) and Aharonov and Kirwan [1]. The inequality for the conformal radii is obtained via an inequality for conformal capacities, which seems to be of independent interest. A number of applications in the theory of functions are discussed. Here we introduce a definition of a class of functions {f}, analytic in the unit disk | §| 0 the set CL (f^) is open (relative to M) and that for 0 0.) We now keep j fixed. Let P1 = (x- y-Jefi . Kt(f.) and x x x a ^ D j f . ( P ) = A , . Denote by Kc(P,) the open d i s k of r a d i u s 6, c e n t e r e d at I>1. If 0 X ("X A ) 4r6 (x -x ) Since this holds for every j, we obtain (under the same assumptions) (1-9) **(x ,A ) ^l*(x ^ ) + [S (x.-x ) ] 1 / . From (1.9) it follows that: (1.10) |f*(P)-f*(P' ) | (i = 1,2), in na, b,(f*) such that: (1.11) |PX-P2 | = 6 k6. Suppose f*(P1) A2 and A + k5 £*(x2,A2). On the other hand inequality (1.9) holds for these values of X T > X 2 > ^ T * ^ 9 ' H e n c e we obtain: 2 > l + t 6 -( i2 ) 2 ] 1 / 2 -* I1"2 > ' which is a contradiction to (1.11). Definition 1.3. Let feB(M) and denote: = {(x5y) |0 < f(x,y) < 1} PI M; YA(f) = {(x,y) |f(x,y) = A} 0 M, 0 < A < 1. A Suppose that f€C(Q(f)). Let PQ = (xQ,y0) be an interior point of Cl(f) and f (PQ) = AQ. We shall say that PQ is a regular point of f, if Sf/5y j4 0 at all the points of the set Y-x PI {x = xQ} and if this set is contained in the interior of fi(f). Otherwise we shall say that PQ is a critical point and AQ a critical value of f on x = x . Lemma 1.2. Let feB(M) n C(n(f)). Suppose that AQ, (0 < 7\Q < 1) , is not a critical value of f on x = x , (0 < xQ < 1) . Then l(x,~k;±) eC in a neighborhood of (xQ,A ) . Proof. Since feB(M), y-v intersects the line x = xn at a 0 finite number of points {p. , . . . ,P.} . Let P. = (x ,y.) and suppose that y, < y2 <...< y,. Then the sequence has alternating signs. Let y. = y.(x,A) be the inverse function (with respect to *j «J y) of A = f(x,y), in a neighborhood of p.. Then for

Radial averaging of domains, estimates for Dirichlet integrals and applications

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