Vector valued minimizers of anisotropic functionals: fractional differentiability and estimate for the singular set

We prove “fractional” higher differentiability for the gradient of minimizers of anisotropic integral functionals, if the growth exponents are no too far apart. This allows us to give an estimate for the Hausdorff dimension of the singular set of the minimizers.     

Lipschitz regularity of the minimizers of autonomous integral functionals with discontinuous non-convex integrands of slow growth

Let be a Borelian function and let (P) be the problem of minimizing

Partial regularity in non-linear elasticity

We prove partial regularity of minimizers of some polyconvex functionals. In particular our results include models such as ∫_Ω a(x,u)(|Du|²+| det Du|²), where a is a bounded H?lder continuous function, such that a(x,u)≥c for some positive constant c. Received: 2 January 2001 / Revised version: 30 August 2001     

A Simple Partial Regularity Proof for Minimizers of Variational Integrals

We consider multi-dimensional variational integrals

Everywhere regularity of functionals with <Emphasis Type="Italic">φ</Emphasis>-growth

We prove C ^1,α -regularity for local minimizers of functionals with φ-growth, giving also the decay estimate. In particular, we present a unified approach in the case of power-type functions. Supported by PRIN Project: “Calcolo delle variazioni e Teoria Geometrica della Misura”.     

Comparison estimates in anisotropic variational problems

A pointwise inequality between the radially decreasing symmetrals of minimizers of (possibly) anisotropic variational problems and the minimizers of suitably symmetrized problems is established. As a consequence, a priori sharp estimates for norms of the relevant minimizers are derived.     

Copolymer–homopolymer blends: global energy minimisation and global energy bounds

We study a variational model for a diblock copolymer–homopolymer blend. The energy functional is a sharp-interface limit of a generalisation of the Ohta–Kawasaki energy. In one dimension, on the real line and on the torus, we prove existence of minimisers of this functional and we describe in complete detail the structure and energy of stationary points. Furthermore we characterise the conditions under which the minimisers may be non-unique. In higher dimensions we construct lower and upper bounds on the energy of minimisers, and explicitly compute the energy of spherically symmetric configurations.     

Partial regularity of minimizers of a functional involving forms and maps

We discuss the partial regularity of minimizers of energy functionals such as

A relaxation result for micromagnetics in SBV

An integral representation for the functional

Partial regularity for a minimum problem with free boundary

Regularity of the free boundary ?{u > 0} of a non-negative minimum u of the functional $\upsilon \mapsto \int\limits_\Omega {\left( {\left| {\nabla \upsilon } \right|^2 + Q^2 \chi _{\left\{ {\upsilon > 0} \right\}} } \right)} $ , where Ω is an open set in ?ⁿ and Q is a strictly positive Hölder-continuous function, is still an open problem for n ≥ 3. By means of a new monotonicity formula we prove that the existence of singularities is equivalent to the existence of an absolute minimum u* such that the graph of u* is a cone with vertex at 0, the free boundary ?{u* > 0} has one and only one singularity, and the set {u* > 0} minimizes the perimeter among all its subsets. This leads to the following partial regularity: there is a maximal dimension k* ≥ 3 such that for n < k* the free boundary ?{u > 0} is locally in Ω a C^1,α-surface, for n = k* the singular set Σ:= ?{u > 0} ? ?_red{u > 0} consists at most of in Ω isolated points, and for n > k* the Hausdorff dimension of the singular set Σ is less than n - k*.     

Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption

We study the limit behaviour of solutions of with initial data k δ ₀ when k → ∞, where h is a positive nondecreasing function and p > 1. If h(r) = r ^β , β > N(p − 1) − 2, we prove that the limit function u _∞ is an explicit very singular solution, while such a solution does not exist if β ≤  N(p − 1) − 2. If lim inf _{r→ 0} r ² ln (1/h(r))  >  0, u _∞ has a persistent singularity at (0, t) (t ≥  0). If , u _∞ has a pointwise singularity localized at (0, 0).     

Regularity of minimizers of W1,p -quasiconvex variational integrals with (p,q)-growth

We consider autonomous integrals

Quasiminimizers in one dimension: integrability of the derivative,inverse function and obstacle problems

It is shown that a K-quasiminimizer u for the one-dimensional p-Dirichlet integral is a K′-quasiminimizer for the q-Dirichlet integral, 1  ≤  q  <  p ₁(p, K), where p ₁(p, K) > p; the exact value for p ₁(p, K) is obtained. The inverse function of a non-constant u is also K′′-quasiminimizer for the s-Dirichlet integral and the range of the exponent s is specified. Connections between quasiminimizers, superminimizers and solutions to obstacle problems are studied.     

On the Schwarz reflection principle for monogenic functions

Let ∑ be either an oriented hyperplane or the unit sphere in , let be open and connected and let be an open and connected domain in such that . If in is a null solution of the Dirac operator (also called a monogenic function in ) which is continuously extendable to , then conditions upon are given enabling the monogenic extension of across . In such a way Schwarz reflection type principles for monogenic functions are established in the Spin (1) and Spin cases. The Spin (1) case includes the classical Schwarz reflection principle for holomorphic functions in the plane. The Spin case deals with so-called “half boundary value problems” for the Dirac operator. Received: 2 February 2006     

Low regularity solution for a nonlocal perturbation of KdV equation

The Cauchy problem for a nonlocal perturbation of KdV equation is considered by the Fourier restriction norm method. Local well-posedness for initial data in and global results for data in are obtained. The second author was supported by the National Natural Science Foundation of China Grant No.10526011.     

A nondegeneracy result for a nonlinear elliptic equation

Let Ω be a smooth bounded domain of with N ≥ 5. In this paper we prove, for ɛ > 0 small, the nondegeneracy of the solution of the problem

Bifurcation problems for superlinear elliptic indefinite equations with exponential growth

This paper deals with the existence and the behaviour of global connected branches of positive solutions of the problem

Optimization by the Homogenization Method for Nonlinear Elliptic Dirichlet Systems

We study the existence of solutions of control problems relative to a nonlinear elliptic system with Dirichlet boundary conditions. In this problem, the control variables are the coefficients of the equations and the open set where they are posed. It is known that this class of problems has no solution in general, but using homogenization results about elliptic systems we show the existence of solutions when the controls are searched in a bigger set. These results are related to the selection of optimal materials and shapes.     

Existence of type (II) regions and convexity and concavity of potential functionals corresponding to jumping nonlinear problems

In this paper we study the jumping nonlinear problem

A new approach to interior regularity of elliptic systems with quadratic Jacobian structure in dimension two

We prove a new regularity result for systems of nonlinear elliptic equations with quadratic Jacobian type nonlinearity in dimension two. Our proof is based on an adaptation of John Lewis’ method which has not been used for such systems so far. Parts of this work have been done while the second and the third author had been enjoying the hospitality of the Department of Mathematics of the University of Pittsburgh. P.H. was supported by NSF grant DMS-0500966. P.S. was partially supported by the MNiSzW grant no 1 PO 3A 005 29. X.Z. was supported by the Academy of Finland, project 207288.