Showing papers in "Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze in 1998"

Global existence and blow-up for a shallow water equation

Abstract: An interesting phenomenon in water channels is the appearance of waves with length much greater than the depth of the water. In 1895 D. J. Korteweg and G. de Vries started the mathematical theory of this phenomenon and derived a model describing unidirectional propagation of waves of the free surface of a shallow layer of water. This is the well-known KdV equation

On the regularity of boundary traces for the wave equation

Abstract: © Scuola Normale Superiore, Pisa, 1998, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Homogenization of elliptic problems in a fiber reinforced structure. non local effects

Abstract: Let S2 be a bounded open subset of]R and (ae) a sequence of functions on S2 taking very high values on a set of 8-periodically distributed fibers of radius r e (r e « ~). We study asymptotically as 8 0 the elliptic equation = f + boundary conditions and find a non local effective equation deduced from a homogenized system of several elliptic equations. Mathematics Subject Classification (1991): 35J20, 73B27. 1. Introduction and statement of the main result Let S2 be a bounded smooth open subset of and p E (1, +oo) . We are concerned with the homogenization of quasilinear elliptic problems well posed in of the form:

Some theoretical results concerning non newtonian fluids of the Oldroyd kind

Abstract: L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Trace inequalities for Carnot-Carathéodory spaces and applications

Abstract: Given a distribution belonging to a sub-elliptic Sobolev space with respect to a system of locally Lipschitz vector fields, we study the problem of its membership to sharp trace spaces with respect to a given Borel measure. Various applications to geometric trace inequalities and to optimal regularity theorems for solutions of quasilinear sub-elliptic equations are presented.

Monotonicity and symmetry of solutions of $p$-Laplace equations, $1 < p < 2$, via the moving plane method

Abstract: In this paper we prove some monotonicity and symmetry properties of positive solutions of the equation - div Du) = f (u) satisfying an homogenuous Dirichlet boundary condition in a bounded domain Q. We assume 1 < p < 2 and f locally Lipschitz continuous and we do not require any hypothesis on the critical set of the solution. In particular we get that if Q is a ball then the solutions are radially symmetric and strictly radially decreasing.

Birational canonical transformations and classical solutions of the sixth Painlevé equation

Abstract: Two topics on the sixth Painleve equation are treated in this paper. In Section 1, a simple construction of a group of birational canonical transformations of the sixth equation isomorphic to the affine Weyl group of D4 root system is given by exploiting an affine Weyl group symmetry of the Hamiltonian structure of the sixth equation defined on the defining variety of the equation. In the rest of this paper (Sections 2-4), based on Umemura’s theory on algebraic differential equations, all one-parameter families of classical solutions of the sixth equation are determined, and the irreducibility of the sixth equation is proved. The latter is a rigorous proof of what Painleve asserted in C. R. Acad. Sci. Paris 143 (1906), 1111-1117.

Rate of Approach to a Singular Steady State in Quasilinear Reaction-Diffusion Equations

Abstract: We study the asymptotic behaviour of global-in-time sQlutions to a quasilinear reaction-diffusion equation in the case when it admits a unique stable stationary solution which is not a bounded function (a singular steady state). We investigate the convergence from below of global solutions to the singular state and discover that such a stabilization is not of a self-similar nature. Actually, it is given by a certain matching of different asymptotic developments in the large outer region closer to the boundary and the thin inner region near the singularity.

Comparison results between minimal barriers and viscosity solutions for geometric evolutions

Abstract: L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Homoclinic and periodic orbits for hamiltonian systems

Abstract: This article deals with the existence of homo clinic and periodic solutions for second order Hamiltonian systems. The main purpose is to consider unbounded potentials which do not satisfy the Ambrosetti-Rabinowitz condition. The method is variational and it combines a perturbation argument with Morse index estimates for minimax critical points.

Viscosity solutions and regularity of the free boundary for the limit of an elliptic two phase singular perturbation problem

Abstract: In this paper we are concerned with the following problem: Study the limit as E 0, of solutions u £ (x ) to the equation: where E > 0 and = 1 p (:~). Here 0 is a Lipschitz continuous function with f3 > 0 in (0,1 ) and 0 0 outside (0,1 ) and j = M. We consider a family US of uniformly bounded solutions to Es in a domain Q C R N and we prove that, under suitable assumptions, the limit function u is a solution to in a pointwise sense at \"regular\" free boundary points and in a viscosity sense. Then, we prove the regularity of the free boundary. In fact, we prove that in the absence of zero phase, if uis nondegenerate at xp E S2 n a {u > 0}, then the free boundary is a surface in a neighborhood of xo. Therefore, u is a classical solution to (E) in that neighborhood. For the general two phase case (which includes, in particular, the one phase case) we prove that, under nondegeneracy assumptions on u, if the free boundary has an inward unit normal in the measure theoretic sense at a point xo n a {u > 0}, then the free boundary is a surface in a neighborhood of xo. Mathematics Subject Classification (1991): 35R35, 35J99, 80A25.

An Exponential Transform and Regularity of Free Boundaries in Two Dimensions

Abstract: We investigate the basic properties of the exponential transform E(z, w) = exp 7r 0 x dA(~) -) ) (z, w E C) of a domain Q c C and compute it in some simple cases. The main result states that if the Cauchy trans- form of the characteristic function of Q has an analytic continuation from C B S2 across aS2 then the same is true for E(z, w), in both variables. If F (z, w) de- notes this analytic-antianalytic continuation it follows that 8 Q is contained in a real analytic set, namely the zero set of F (z, z). This gives a new approach to the regularity theory for free boundaries in two dimensions.

Exact boundary controllability of Galerkin's approximations of Navier-Stokes equations

Abstract: We consider the 2-d and 3-d Navier - Stokes equations in a bounded smooth domain with a boundary control acting on the system through the Navier slip boundary conditions. We introduce a finite-dimensional Galerkin approximation of this system. Under suitable assumptions on the Galerkin basis we prove that this Galerkin approximation is exactly controllable. Moreover we prove that the cost of controlling is independent of the presence of the nonlinearity on the system. Our assumptions on the Galerkin basis are related to the linear independence of suitable traces of its elements over the boundary. At this respect, the one-dimensional Burgers equation provides a particularly degenerate example that we study in detail. In this case we prove local controllability results.

The stationary Boltzmann equation in the slab with given weighted mass for hard and soft forces

Abstract: The stationary Boltzmann equation for hard and soft forces is considered in the slab. An L 1 existence theorem is proven in a given indata context with fixed total weighted mass. In the proof a new direct approach is introduced, which uses a certain coupling between mass and boundary flow. Compactness properties are extracted from entropy production estimates and from the boundary behaviour.

A class of nonlinear conservative elliptic equations in cylinders

Abstract: Let (M, g) be a compact n-dimensional manifold without boundary and Ag the Laplace-Beltrami operator on M. This paper studies the asymptotic properties of the following conservative system (S)utt + Ogu + uq Àu = 0 on R+ x M and their links with the homogeneous solutions of (S). 1. Introduction The study of asymptotics of the following class of conformally invariant Emden-Fowler equations in JRN 101 gives rise to the following nonlinear equation in (201300, oo) x SN-1, where OSN-1 is the Laplace-Beltrami operator on the unit sphere SN-1 of JRN, via the following classical change of variable One of the main feature of this equation is the conservation of energy (equivalent to Pohozaev’s identity): Pervenuto alla Redazione il 8 novembre 1996 e in forma definitiva il 2 luglio 1997.. 250 As a result of the works of Obata [0b] and Caffarelli-Gidas-Spruck [CGS], the asymptotic behaviour of the solutions of (1.2) as well as the global solutions are now well understood in the case when c = 0, but it is important to notice that this understanding mainly comes from the equation (1.1) itself and not from the study of (1.2): the main point is that the solutions behave asymptoticaly like the solutions of the associated O.D.E. It appears that when c is not 0, nothing is known except in the radial case where the relation (1.4) plays a crucial role: in particular there may exist solutions of (1.2) under the form where A is a skew symmetric matrix. The purpose of this paper is to extend this type of problem to a more general setting by considering the following equation in [0,00) x M where (M, g) be a n-dimensional compact Riemannian manifold without boundary, Ag the Laplacian on M and q and k are constant, q > 1. Let us first study the stationnary equation associated to (1.6), that is and in particular look under what conditions all the (positive) solutions of (1.7) are constant (by a solution we always mean a C2 (M)-function). Let À1 denote the first nonzero eigenvalue of -Ag, then two types of results are obtained in that direction. The first one points out the role of the curvature tensor and in particular its trace, the Ricci tensor: THEOREM 2.1. Assume that the Ricci tensor Riccg of g satisfies for some nonnegative R, that ~, is nonnegative and

Gradient flow for the one-dimensional Mumford-Shah functional

Abstract: In order to introduce a notion of gradient flow for the one-dimensional Mumford-Shah functional MS(u), we consider a family of regular functionals, defined in spaces of piecewise constant functions, which converge in a variational sense to MS(u). Moreover, given an initial datum uo, with MS(uo) +00, and a family } of piecewise constant approximations of uo, we consider the evolution problems We show that for large classes of initial data, the family (us (t) ) converges, as 8 0+, to a certain u (t), which is the solution of the heat equation with homogeneous Neumann boundary conditions in a suitable variable domain. On the other hand, we show that, for some special uo, the family has infinitely many limit points as 8 ~ 0+. Mathematics Subject Classification (1991): 58D25 (primary), 34G20 (secondary). 1. Introduction In last years many variational problems with free discontinuities have been studied. The canonical examples are the minimum problems related to the so called Mumford-Shah functional, defined by where Q is an open subset of R’, u belongs to the space of special functions with bounded variation (see Section 2), Vu is the approximate gradient

Lindstedt series, ultraviolet divergences and Moser's theorem

Abstract: Moser’s invariant tori for a class of nonanalytic quasi integrable even Hamiltonian systems are shown to be analytic in the perturbation parameter. We do so by exhibiting a summation rule for the divergent series ("Lindstedt series") that formally define them. We find additional cancellations taking place in the formal series, besides the ones already known and necessary in the analytic case (i.e. to prove convergence of Lindtsedt algorithm for Kolmogorov’s invariant tori). The method is interpreted in terms of a non renormalizable quantum field theory, considerably more singular than the one we pointed out in the analytic case.

Strong uniqueness for certain infinite dimensional Dirichlet operators and applications to stochastic quantization

Abstract: Strong and Markov uniqueness problems in L2 for Dirichlet operators on rigged Hilbert spaces are studied. An analytic approach based on a priori estimates is used. The extension of the problem to the Lp-setting is discussed. As a direct application essential self-adjointness and strong uniqueness in Lp is proved for the generator (with initial domain the bounded smooth cylinder functions) of the stochastic quantization process for Euclidean quantum field theory in finite volume

On Liouville theorem and apriori estimates for the scalar curvature equations

Abstract: In this article, we study the problem of nonexistence of positive solutions of the equation where K(x) is a C homogeneous function of degree 1 > 0. Suppose that K (x ) satisfies the nondegenerate condition, for x E JRn B {OJ, where ci are positive constants. We prove that equation ( 1 ) admits no positive solutions. This Liouville Theorem allows us to derive apriori estimates for positive solutions of scalar curvature equations in the region where the scalar curvature function is nonpositive. Various apriori estimates are also derived under different circumstances. Mathematics Subject Classification (1991): 35J60, 45G10. 1. Introduction Let (M, go) be a Riemannian manifold of dimension n > 3. For a given smooth function R on M, one would like to find a metric g conformal to 4 go such that R is the scalar curvature of the new metric g. Set g V n-2 go for some positive function v, then the question above is equivalent to finding positive solutions of where Ogo is the Beltrami-Laplace operator of (M, go) and k is the scalar curvature of go. In recent years, there have been a lot of progress in understanding equation ( 1.1 ), in particular, when (M, go) is the standard n-dimensional Pervenuto alla Redazione il 2 febbraio 1998 e in forma definitiva il 19 agosto 1998. 108 sphere Sn . In this case, k (x ) = n (n 1) and then equation (1.1) becomes By using the stereographic projection 7r of S’ onto M" and letting u (x) _ o "20132 with x = 7r(y) for y E S’ and for some suitable positive constant Cn, equation (1.2) reduces into where K (x ) = (x ) ) for x E In general, if (M, go) is a locally conformally flat manifold, a local flat metric can be chosen and then, equation ( 1.1 ) is reduced to equation (1.3) in an open set of R’~. It is well-known that, when R (y) is a positive constant identically for y E S’~, equation (1.2) possesses a family of solutions, whose total energy could be concentrated in a small neighborhood of some point yo E S’~. Thus, it is of great interest from the viewpoint of PDE to study the blow-up behavior for a sequence of positive solutions of (1.2) when R (y) is a nonconstant function. For works in this aspect, we refer the reader to [3], [6,7], [11,12], [20] and references therein. To do the blow up analysis, we first rescale solutions near a blow-up point. Since the solution structure of equation (1.3) was completely understood for the case K (x) =a positive constant, it is easier to describe the limit of the rescaling solutions when As) is a positive function. This is the reason why most works have only considered the situation where is positive. In this paper, we will consider the problem of finding apriori estimates for the region where the curvature function R is nonpositive. In [5], Chen-Li studied equation (1.2) on S’~ and proved some apriori bound for solutions in the region {y E S’ I R(y) s 0} by assuming 0 whenever R (y) = 0. However, their argument seems to work only for solutions globally defined on the whole space In this paper, we want to extend their result for solutions defined locally, which can be applied in the general case when M is locally conformally flat. We believe that the result in local nature should be more useful. For applications, we can prove the apriori bound without the condition B1 R (y) i= 0 whenever R (y) 0, although a nondegenerate condition is still required. It is quite well-known that the establishment of apriori bounds is closely related to the Liouville theorem. The Liouville theorem always plays an important role in the theory of elliptic equations. For the scalar curvature equations (1.3) in R , we [14] have proved the Liouville theorem for several classes of functions K. One of them is that depends only on the x, variable only. Assume K (xl ) is nondecreasing in xl, K = ~i 1 for xl b and K n K2 > max(0, for a, we [14] proved that equation (1.3) possesses no positive solutions in The Liouville theorem for this class of K was first observed by Y.Y. Li [10], where he proved that there are no positive solutions / 109 with the decay rate O(lxI2-n) at infinity. In fact, he asked whether equation (1.3) possesses no positive solutions when a nonconstant K (xl , ... , xn) is bounded by two positive constants and K (xl, ... , xn) is nondecreasing in xi. In general, this problem still remains open. Our first result is: THEOREM 1. 1. Let K (xl , X2, ... , Xn) x’ for some positive integer m. Then equation (1.3) possesses no positive solutions in JRn. Obviously, K (xl ) is increasing in Xl only when m is an odd positive integer. For the case of even positive integer, Theorem 1.1 can be extended to: THEOREM 1.2. Suppose that K > 0 in and satisfies K (t~) = tl K (ç) for t > 0 I = 1, where 1 > 0 is a constant. Assume there exists an open set c~o C Sn such that K (~ ) > co > 0 for some constant co and ~ E Then equation ( 1.3) possesses no positive solutions. In fact, for a homogeneous function K, we have more general result than Theorem 1.1. Note that in the following theorem, K could allow to change signs. THEOREM 1. 3. Suppose that K is a C homogeneous function of degree 1 > 0 and satisfies for positive constants Ci and C2. Then equation (1.3) possesses no positive solutions in JRn. As mentioned before, the Liouville theorem is closely related to the existence of apriori bound of solutions. An immediate consequence of Theorem 1.1 is the following theorem. (Throughout the paper, Br always denotes the open ball with center 0 and the radius r). COROLLARY 1.4. Suppose satisfies and for x E B1. Then there exists a constant c which depends on C1, C2 and the dimension n such that holds for any solution u of 110 DEFINITION. A function K E C 1 (B1 ) with K (0) = 0 is said to satisfy the nondegenerate condition at 0 if in a neighborhood of 0, I~ can be written by COROLLARY 1.5. Suppose K E C1(R1) with K (0) = 0 satisfies [NG] at 0. Let ui be a sequence of solutions of (1.6). Assume that ui is uniformly bounded in any compact set of BIB101. Then ui is uniformly bounded in Bl. DEFINITION. A function E is said to satisfy condition [M] if (i) K > 0 in B I and 7~ 0 whenever K (x) # 0. (ii) The zero set A = f x E B1 I K (x) = 0} is a k-dimensional submanifold with 0 k n. Let N be a tube neighborhood of A and let yr denote the orthorgonal projection from N onto A such = d(x), where d(x) = d(x, A) is the distance of x to A. For x E N, K satisfies where is a positive homogeneous function of degree 1 in (TxoA)1-, the orthogonal complement of the tangent subspace of A at xo. It is easy to see that notions of [NG] and [M] can be defined in a locally confomally flat manifold, and their definitions are independent of the choice of the local flat metrics. COROLLARY 1.6. with K (0) = 0 and satisfies the assumption [M]. Then there exists a constant c such that holds for any solution of ( 1.6). Together with Corollary 1.4 through Corollary 1.6, we have the following apriori estimates for equation (1.1). To state our result, we let SZ_ lpo E M I 0 in a neighborhood of po) and QO _ {po E M ~ R (p°) = 0 and there exists a neighborhood U of po such that R is nonnegative in U, but does not vanish identitically in U}. Obviously, SZis an open subset in M, and SZis disjoint from QO. 111 THEOREM 1.7. Let (M, go) be a locally conformally flat manifold and R be a C 1 function on M. Assume that (i) IfQo i= 0, then R has only a finite set of critical points lpi, ... , p,, I on and satisfies the nondegenerate condition at each pj, and (ii) If 0, then S2° is a union of a finite number of submanifold and R satisfies [M] on each component of Qo. Then there exists 80 > 0 and c > 0 such that if u is a positive solution of ( 1.1 ), then for any p E M and R ( p) 80. We note that if a C 1 function R has no critical points with vanishing critical values, then R satisfies the assumption of Theorem 1.7. In this case, Theorem 1.7 was previously proved in [5] when (M, go) is the standard ndimensional sphere. However, Theorem 1.7 is much stronger than the one in [5] even when (M, go) is the n-dimensional sphere. For example, if at each critical point p of R with R (p) 0, the Hessian of R at p is non-singular, then R satisfies the assumption of Theorem 1.7. Thus, we extend the previous result of [5] to allow R to have critical points with zero critical value, of course, the nondegenerate condition is still needed. As mentioned before, the apriori bound for the part where R is positive was obtained in a number of recent works, e.g., see [3], [6,7], [10,11] and references therein. Together with Theorem 1.7, one could obtain apriori bound for solutions for some class of R where R is not assumed to be positive. For example, we have the following result on S3. THEOREM 1.8. Let (S3, go) be the standard 3-sphere and R be a given Morse function on S3. Assume R satisfies 0 whenever R (y) > 0 and p R (y) = 0. Then there exists a constant c > 0 such that for any solution v of (1.2), one has Furthermore, the Leray-Schauder degree d for equation (1.2) is given by where A+ = {p e S3 p is a critical point o, f’ R with R(p) > 0 and 0}, the Morse index o, f R at We note that the Leray-Schauder degree d is defined as the standard topological degree of the nonlinear map v -~ g ( 00 4 ) -1 R v5 from c2,a (S3) to itself for 0 a 1. For reference, please see [15]. For a locally conformally flat manifold which is not conformally equivalent to we can apply the positive mass theorem and obtain: 112 THEOREM 1.9. Let (M, go) be a locally conformally flat n-manifold and R be a Cn-2 function which for n > 4, in addition assume that for any 8 > 0, there exists a neighborhood U of the critical set f p E M pR ( p) = 0 and R ( p) > 01 such that where 1 = n 2. Suppose R also satisfies the assumption of Theorem 1.7 on the and S2° if they are not empty. Then there exists a positive constant c > 0 such that for p E M and for any solution v of ( 1.1 ). Furthermore, if the first eigenvalue of the conformal Laplacian operator is positive and maxM R > 0. Then t

On the Schauder estimates of solutions to parabolic equations

Abstract: In this paper we give a priori estimates on asymptotic polynomials of solutions to parabolic differential equations at any points. This leads to a pointwise version of Schauder estimates. The result in this paper improves the classical Schauder estimates in a way that the estimates of solutions and their derivatives at one point depend on the coefficient and nonhomogeneous terms at this particular point.

On the Birkhoff normal form of a completely integrable hamiltonian system near a fixed point with resonance

Abstract: We consider an integrable Hamiltonian system with a real analytic Hamiltonian H near an elliptic fixed point P. If H has a simple resonance and admits a semisimple Hessian at P we show that there exists a real analytic change of coordinates which brings the Hamiltonian into normal form. In the new coordinates, the level sets of the system are analyzed in terms of the nature of the simple resonance.

Global solutions of the Cauchy problem for a viscous polytropic ideal gas

Abstract: We consider the Cauchy problem for a viscous polytropic ideal gas in R’ (n = 2 or 3). First we derive an a priori estimate for (smooth) solutions for small eo which may be used to show the existence of weak solutions, then we prove the existence and uniqueness of global (smooth) solutions for small Eo, where eo and Eo, depending on dimensions, are bounded from above by the Sobolev norms of the initial data. (In two dimensions eo and Eo are bounded from above by the 1 x H 1-norms of (po p, vo, 00 6) respectively, where po, vo and 00 are the initial density, the initial velocity, and the initial temperature respectively, a E (2, oo), > 0 are constants.) 1. Introduction The motion of a viscous polytropic ideal gas in (n = 2 or 3) is described by the following equations in Eulerian coordinates (cf. [3], [26], [2]) Here P, 0, and v = (VI, ... , vn) are the density, the absolute temperature and the velocity respectively, R, cv and K are positive constants; X and it are the constant viscosity coefficients, ti > 0, À + 2~c,~/n > 0; D = D(v) is the deformation tensor B

Inegalites d'auto-intersection pour les courants positifs fermes definis dans les varietes projectives

Abstract: Le premier objectif de cet article est d’etablir, etant donne un courant positif ferme defini sur une variete projective, des inegalites d’auto-intersection qui permettent de borner le degre des strates ou la multiplicite est constante. A l’aide du theoreme de plongement de Matsusaka, on se ramene au cas de l’espace projectif et on utilise alors un potentiel de Skoda qu’on envisage ici d’un point de vue geometrique. L’un des points essentiels est que la methode utilisee permet de traiter le cas d’un courant de dimension quelconque. Seul le cas de la codimension 1 etait connu auparavant. Dans la suite de l’article, on calcule explicitement le noyau dans l’espace projectif definissant le potentiel utilise, afin de bien faire le lien entre ses deux constructions. On y arrive en remontant a l’espace affine. On propose ensuite aussi une methode intrinseque plus naturelle, qui relie ce calcul a la resolution explicite de l’equation d’Euler- Green. Cette derniere est classique en theories de Nevanlinna et d’Arakelov et a deja ete resolue par Bismut-Bost-Gillet-Soule. Dans cet article on donne egalement une presentation self-contained d’une solution elementaire de cette equation, obtenue a partir de la formule de King.

On one-sided BMO and Lipschitz functions

Abstract: Two basic results concerning functions with conditions on the mean oscillation are extended to the one-sided setting: the estimate of its distance to L °° and the pointwise one-sided regularity. In addition we investigate the structure of these one-sided regular functions and their basic properties.

The dam problem for nonlinear Darcy's laws and Dirichlet boundary conditions.

Abstract: The subject of this paper is the study of a free boundary problem for a steady fluid flow through a porous medium, in which the classical Darcy law (1) −!v = ar(p(x)+xn), x = (x1, · · · , xn) 2 Rn, a > 0, is replaced by the nonlinear law (2) |−!v |m−1−!v = ar(p(x)+xn), x = (x1, · · · , xn) 2 Rn, a, m > 0, where −!v and p are, respectively, the velocity and the pressure of the fluid. This approach is particularly interesting because Darcy’s law was established on a purely experimental basis; but it is not clear why, from a physical point of view, the specific form of (2) gives a better model for the dam problem. The authors first reduce the problem to a variational inequality involving the degenerate Laplacian operator for the hydrostatic head u(x) = p(x) + xn. Then,using a perturbation argument, they prove existence of weak solutions. The remaining portion of the paper is devoted to the study of the qualitative properties of the solutions. In particular, it is proven that the free boundary is a lower semicontinuous curve of the form xn = (x1, · · · , xn−1),and that there is a unique minimal solution. Moreover, in the two-dimensional case the authors show that is actually continuous, and that there is a unique S3-connected solution.

Isometries of the Teichmüller metric

Abstract: We study the curvature properties of the Kobayashi-Teichriiuller metric showing in particular that the holomorphic curvature is constant -4. Carrying a program due to Royden, we describe the consequences on complex geodesics. The results are applied to characterize biholomorphic maps into Teichmfller spaces in finite and infinite dimension.

Optimal regularity for mixed parabolic problems in spaces of functions which are Hölder continuous with respect to space variables

Abstract: We give necessary and sufficient conditions in order that a general linear mixed parabolic problem have a solution in spaces of functions with derivatives which are Holder continuous with respect to space variables. Autonomous and nonautonomous problems are considered.