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     

Approximating <Emphasis Type="Italic">J</Emphasis>-holomorphic curves by holomorphic ones

Given an almost complex structure J in a cylinder of (p > 1) together with a compatible symplectic form and given an arbitrary J-holomorphic curve without boundary in that cylinder, we construct an holomorphic perturbation of , for the canonical complex structure J ₀ of , such that the distance between these two curves in W ^1,2 and norms, in a sub-cylinder, are controled by quantities depending on J, and by the area of only. These estimates depend neither on the topology nor on the conformal class of . They are key tools in the recent proof of the regularity of 1-1 integral currents in [RT].Received: 2 October 2003, Accepted: 18 November 2003, Published online: 25 February 2004     

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     

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     

The Nehari manifold for a semilinear elliptic equation involving a sublinear term

The Nehari manifold for the equation for together with Dirichlet boundary conditions is investigated in the case where . Exploiting the relationship between the Nehari manifold and fibrering maps (i.e., maps of the form where J is the Euler functional associated with the equation), we discuss how the Nehari manifold changes as changes and show how this is linked to results on bifurcation from infinity which are associated with the problem.Received: 8 December 2003, Accepted: 10 May 2004, Published online: 16 July 2004Mathematics Subject Classification (2000): 35J20, 35J25     

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     

On the isoperimetric inequality on a minimal surface

We derive an explicit formula for the isoperimetric defect of an arbitrary minimal surface ,in terms of a double integral over the surface of certain geometric quantities, together with a double boundary integral which always has the correct sign. As a by-product of these computations we show that the best known universal isoperimetric estimate, that for any minimal surface (due to L. Simon), may be improved to the universal estimate .Received: 21 June 2001, Accepted: 16 June 2002, Published online: 5 September 2002     

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.     

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     

$C^\infty$ 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

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     

Resonance Free Domains for Non Globally Analytic Potentials

We study the resonances of the semiclassical Schr?dinger operator near a non-trapping energy level in the case when the potential V is not necessarily analytic on all of but only outside some compact set. Then we prove that for some and for any C > 0, P admits no resonance in the domain if V is , and if V is Gevrey with index s. Here does not depend on h and the results are uniform with respect to h > 0 small enough. Submitted 05/02/02, accepted 06/05/02 An erratum to this article is available at .     

Biharmonic map heat flow into manifolds of nonpositive curvature

Let M ^m and N be two compact Riemannian manifolds without boundary. We consider the L ² gradient flow for the energy . If and N has nonpositive sectional curvature we show that the biharmonic map heat flow exists for all time, and that the solution subconverges to a smooth harmonic map as time goes to infinity. This reproves the celebrated theorem of Eells and Sampson [6] on the existence of harmonic maps in homotopy classes for domain manifolds with dimension less than or equal to 4.Received: 27 March 2003, Accepted: 5 April 2004, Published online: 16 July 2004Mathematics Subject Classification (2000): 58E20, 58J35     

Stationary measures and rectifiability

For integers , we consider -valued Radon measures on an open set which satisfy

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

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     

Infinitely many bound states for some nonlinear scalar field equations

In this paper we consider the problem in , where p > 2 and if N > 2. Assuming that the potential a(x) is a regular function such that 0$" align="middle" border="0"> and that verifies suitable decay assumptions, but not requiring any symmetry property on it, we prove that the problem has infinitely many solutions.Received: 3 December 2003, Accepted: 10 May 2004, Published online: 22 December 2004     

A chain rule in $L^{1}\left({\operatorname*{div};\Omega}\right)$ and its applications to lower semicontinuity

A chain rule in the space is obtained under weak regularity conditions. This chain rule has important applications in the study of lower semicontinuity problems for general functionals of the form with respect to strong convergence in . Classical results of Serrin and of De Giorgi, Buttazzo and Dal Maso are extended and generalized.Received: 1 June 2002, Accepted: 7 January 2003, Published online: 6 June 2003     

On the \Gamma-limit of the Mumford-Shah functional

We study by means of -convergence the asymptotics of the rescaled Mumford-Shah functional