Boundary blow-up for a Brezis-Peletier problem on a singular domain

Let , , be a bounded domain as defined by Flucher, Garroni and Müller [6], which has a singular point such that the Robins function achieves its infimum at . Considering the elliptic problem in ; u = 0 on , with p = (N + 2)/(N-2), , and a minimizing solution of , concentrates at as goes to zero.Received: 15 September 2002, Accepted: 5 November 2002, Published online: 16 May 2003Mathematics Subject Classification: 35J65Angela Pistoia: The author is supported by M.U.R.S.T., project Metodi variazionali e topologici nello studio di fenomeni non lineari     

On the relaxation and the Lavrentieff phenomenon for variational integrals with pointwise measurable gradient constraints

Relaxation problems for a functional of the type are analyzed, where is a bounded smooth open subset of and g is a Carathéodory function. The admissible functions u are forced to satisfy a pointwise gradient constraint of the type for a.e. being, for every , a bounded convex subset of , in general varying with x not necessarily in a smooth way. The relaxed functionals and of G obtained letting u vary respectively in , the set of the piecewise C ¹-functions in , and in in the definition of G are considered. For both of them integral representation results are proved, with an explicit representation formula for the density of . Examples are proposed showing that in general the two densities are different, and that the one of is not obtained from g simply by convexification arguments. Eventually, the results are discussed in the framework of Lavrentieff phenomenon, showing by means of an example that deep differences occur between and . Results in more general settings are also obtained.Received: 18 December 2002, Accepted: 18 November 2003, Published online: 16 July 2004Mathematics Subject Classification (2000): 49J45, 49J10, 49J53This work is part of the European Research Training Network Homogenization and Multiple Scales (HMS 2000), under contract HPRN-2000-00109. It is also part of the 2003-G.N.A.M.P.A. Project Metodi Variazionali per Strutture Sottili, Frontiere Oscillanti ed Energie Vincolate.     

Characterization of ideal knots

We present a characterization of ideal knots, i.e., of closed knotted curves of prescribed thickness with minimal length, where we use the notion of global curvature for the definition of thickness. We show with variational methods that for an ideal knot , the normal vector at a curve point is given by the integral over all vectors against a Radon measure, where realizes the given thickness. As geometric consequences we obtain in particular, that points without contact lie on straight segments of , and for points with exactly one contact point we have that points exactly into the direction of Moreover, isolated contact points lie on straight segments of , and curved arcs of consist of contact points only, all realizing the prescribed thickness with constant (maximal) global curvature.Received: 1 January 2003, Accepted: 12 March 2003, Published online: 1 July 2003Mathematics Subject Classification (2000): 53A04, 57M25, 74K05, 74M15, 92C40     

A new example of singular height transition for capillary surfaces

Finn and Kosmodemyanskii, Jr. gave an example of a domain containing a disk , and of a family of domains converging to as , such that the heights u ^t of capillary surfaces in vertical tubes with the sections in a gravity field g satisfy for every , but for which u ¹< u ⁰ over for all g > 0. In subsequent work, Finn and Lee characterized the most general convex that leads to such a discontinuous transition when is a disk. It has been suggested that the cause for this curious behavior is related to the fact that in all cases considered the boundaries of the have a discontinuity in their curvatures, that is bounded below in magnitude. In the present note we present an alternative form of the example, in which the domains are disks concentric to . Thus, the limited smoothness in the original example of the convergence to of the approxim ating domains cannot be viewed as the root cause of the anomaly. The procedure presented here leads to explicit bounds, which were not available in the earlier forms of the example.Received: 3 September 2002, Accepted: 17 February 2003, Published online: 1 July 2003Mathematics Subject Classification: 76B45, 53A10, 49Q10     

Regularity and relaxed problems of minimizing biharmonic maps into spheres

For and , we show that any minimizing biharmonic map from to S^k is smooth off a closed set whose Hausdorff dimension is at most n-5. When n = 5 and k = 4, for a parameter we introduce a -relaxed energy of the Hessian energy for maps in so that each minimizer of is also a biharmonic map. We also establish the existence and partial regularity of a minimizer of for .Received: 5 April 2004, Accepted: 19 October 2004, Published online: 10 December 2004     

Minimal nodal solutions of the pure critical exponent problem on a symmetric domain

We establish existence of nodal solutions to the pure critical exponent problem in u = 0 on where a bounded smooth domain which is invariant under an orthogonal involution of We extend previous results for positive solutions due to Coron, Dancer, Ding, and Passaseo to existence and multiplicity results for solutions which change sign exactly once.Received: 4 April 2003, Accepted: 26 August 2003, Published online: 24 November 2003Mathematics Subject Classification (2000): 35J65, 35J20Research partially supported by PAPIIT, UNAM, under grant IN110902-3.     

A new approach to Sobolev spaces and connections to $\mathbf\Gamma$-convergence

This is a follow-up of a paper of Bourgain, Brezis and Mironescu [2]. We study how the existence of the limit

Compensated compactness for nonlinear homogenization and reduction of dimension

We study the limit behaviour of some nonlinear monotone equations, such as: , in a domain which is thin in some directions (e.g. is a plate or a thin cylinder). After rescaling to a fixed domain , the above equation is transformed into: , with convenient operators and . Assuming that and the inverse of have particular forms and satisfy suitable compensated compactness assumptions, we prove a closure result, that is we prove that the limit problem has the same form. This applies in particular to the limit behaviour of nonlinear monotone equations in laminated plates.Received: 16 October 2002, Accepted: 12 June 2003, Published online: 22 September 2003Mathematics Subject Classification (2000): 35B27, 35B40, 74Q15     

Higher-order energy expansions and spike locations

We consider the following singularly perturbed semilinear elliptic problem: where is a bounded domain in R ^N with smooth boundary , is a small constant and f is some superlinear but subcritical nonlinearity. Associated with (I) is the energy functional defined by where . Ni and Takagi ([29, 30]) proved that for a single boundary spike solution , the following asymptotic expansion holds: where c ₁ > 0 is a generic constant, is the unique local maximum point of and is the boundary mean curvature function at . In this paper, we obtain a higher-order expansion of where c ₂, c ₃ are generic constants and is the scalar curvature at . In particular c ₃ > 0. Some applications of this expansion are given.Received: 14 January 2003, Accepted: 28 July 2003, Published online: 15 October 2003Mathematics Subject Classification (2000): Primary 35B40, 35B45; Secondary 35J25     

A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator

We prove a Sobolev inequality with remainder term for the imbedding , arbitrary, generalizing a corresponding result of Bianchi and Egnell for the case m = 1. We also show that the manifold of least energy solutions of the equation is a nondegenerate critical manifold for the corresponding variational integral. Finally we generalize the results of J. M. Coron on the existence of solutions of equations with critical exponent on domains with nontrivial topology to the biharmonic operator.Received: 21 March 2002, Accepted: 5 November 2002, Published online: 16 May 2003     

The obstacle problem for Monge Ampere equation

We consider the following obstacle problem for Monge-Ampere equation and discuss the regularity of the free boundary . We prove that is if f is bounded away from 0 and , and it is C ^1,1 if .Received: 4 February 2003, Accepted: 3 March 2004, Published online: 16 July 2004     

Formation of cracks under deformations with finite energy

With a map , , that belongs to the John Ball class where n-1 < p < n and one can associate a set valued map F whose values are subsets of describing the topological character of the singularity of f at . verak conjectured that , where S is the set of points at which f is not continuous and is the Hausdorff measure. The purpose of our paper is to confirm this expectation.Received: 3 March 2003, Accepted: 15 April 2003, Published online: 1 July 2003Mathematics Subject Classification (2000): 74B20Piotr Hajasz: Hajasz was supported by KBN grant 2 PO3A 028 22 and Koskela by the Academy of Finland,SA-34082. Part of this research was done while Hajasz was at the Department of Mathematics of the University of Michigan. He wishes to thank UM for the support and hospitality.     

$C^\infty$ regularity of the free boundary for a two-dimensional optimal compliance problem

We study the regularizing effect of perimeter penalties for a problem of optimal compliance in two dimensions. In particular, we consider minimizers of

A sharp attainment result for nonconvex variational problems

We consider the problem of minimizing autonomous, multiple integrals like

Singular limits in Liouville-type equations

We consider the boundary value problem in a bounded, smooth domain in with homogeneous Dirichlet boundary conditions. Here 0,k(x) $$ " align="middle" border="0"> is a non-negative, not identically zero function. We find conditions under which there exists a solution which blows up at exactly m points as and satisfies . In particular, we find that if , 0 $" align="middle" border="0"> and is not simply connected then such a solution exists for any given Received: 11 February 2004, Accepted: 17 August 2004, Published online: 22 December 2004     

On a singular behavior of capillary surfaces

Mario Miranda raised the question, whether a capillary tube of section in an infinite bath under a uniform downward gravity field always raises liquid higher over its section than does a tube of the same material and section . Finn and Kosmodemyanskii, Jr. recently gave conditions under which a negative answer is obtained for all gravity small enough; they gave also an example in which is a disk and in which the answer changes in a discontinuous way, for fixed and differentiable change of . In the present work we characterize the most general convex for which such a discontinuous change can occur.Received: 3 September 2002, Accepted: 23 January 2003, Published online: 1 July 2003Mathematics Subject Classification: 76B45, S3A10, 49Q10     

On nonlocal calculation for inhomogeneous indefinite Neumann boundary value problems

This paper concerns with a family of inhomogeneous Neumann boundary value problems having indefinite nonlinearities which depend on a real parameter . We discuss the existence and the multiplicity of positive solutions with respect to . Developing the fibering method further, we can introduce a constructive concept of the calculation of certain nonlocal intervals , the so-called sufficient intervals of the existence. Then we are able to prove some new results on the existence and the multiplicity of positive solutions for .Received: 22 December 2003, Accepted: 29 January 2004, Published online: 16 July 2004Mathematics Subject Classification (2000): 35J70, 35J65, 47H17     

Improved regularity of harmonic map flows with Hölder continuous energy

For a smooth harmonic map flow with blow-up as , it has been asked [5,6,7] whether the weak limit is continuous. Recently, in [12], we showed that in general it need not be. Meanwhile, the energy function , being weakly positive, smooth and weakly decreasing, has a continuous extension to [0,T]. Here we show that if this extension is also Hölder continuous, then the weak limit u(T) must also be Hölder continuous.Received: 1 September 2003, Accepted: 7 October 2003, Published online: 25 February 2004Version of 19/9/03. Partly supported by an EPSRC Advanced Research Fellowship.     

Some improved Caffarelli-Kohn-Nirenberg inequalities

For 1 < p < N and we obtain the following improved Hardy-Sobolev Inequalities where 1 < q < p and if , if , for some positive constant . Also we give an alternative proof of the optimal improved inequality for p = 2 by Wang-Willem in [16]. Received: 2 February 2004, Accepted: 12 July 2004, Published online: 3 September 2004 Mathematics Subject Classification (2000): 35J20, 35P05, 35R05, 46E30, 46E35 Partially supported by Project BFM2001-0183     

The Dirichlet energy of mappings from ${\bf B^3}$ into a manifold: density results and gap phenomenon

Weak limits of graphs of smooth maps with equibounded Dirichlet integral give rise to elements of the space . We assume that the 2-homology group of has no torsion and that the Hurewicz homomorphism is injective. Then, in dimension n = 3, we prove that every element T in , which has no singular vertical part, can be approximated weakly in the sense of currents by a sequence of smooth graphs {u _k } with Dirichlet energies converging to the energy of T. We also show that the injectivity hypothesis on the Hurewicz map cannot be removed. We finally show that a similar topological obstruction on the target manifold holds for the approximation problem of the area functional.Received: 9 May 2003, Accepted: 5 June 2003, Published online: 25 February 2004