New regularity theorems for non-autonomous variational integrals with (p, q)-growth

One prominent problem in the calculus of variations is minimizing anisotropic integrals with a (p, q)-elliptic density F depending on the gradient of a function ${w : \Omega \rightarrow \mathbb{R}^N}$ with ${\Omega \subset \mathbb{R}^n}$ . The best known sufficient bound for regularity of solutions is q <?p (n?+?2)/n. On the other hand, if we allow an additional x-dependence of the density we have the much weaker result q <?p (n + 1)/n. If one additionally imposes the local boundedness of the minimizer, then these bounds can be improved to q <?p?+?2 and q <?p?+?1. In this paper we give natural assumptions for F closing the gap between the autonomous and non-autonomous situation.     

Regularity results for minimizers of (2, q)-growth functionals in the Heisenberg Group

We consider integral functionals in the Heisenberg group, whose convex C ²-integrand has quadratic growth from below, and growth of order q > 2 from above. We prove Hölder regularity for the full gradient of minimizers under the condition that q is less than an explicitly calculated dimension-dependent bound.     

Partial regularity for minima of higher order functionals with p(x)-growth

For higher order functionals $\int_\Omega f(x, \delta u(x), {D^m}u(x))\,dxFor higher order functionals with p(x)-growth with respect to the variable containing D ^m u, we prove that D ^m u is H?lder continuous on an open subset of full Lebesgue-measure, provided that the exponent function itself is H?lder continuous.     

The theorem on convergence with a functional for integral functionals with p(x)- and p(x, u)-growth

We continue studying weak convergence for the integral functionals satisfying p(x)- and p(x, u)-growth conditions. We obtain the theorem on convergence with a functional and some results on the relation between integral functionals and their abstract lower semicontinuous extensions.     

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

We consider autonomous integrals

Lower closure for orientor fields by lower semicontinuity of outer integral functionals

Summary A classical lower semicontinuity result in optimal control theory and the calculus of variations can be extended to outer integral functionals (viz. integral functionals with nonmeasurable integrands). As a consequence, measurability of the Lagrangian does not have to be guaranteed anymore when applying the deparametrization procedure to existence problems in optimal control theory.     

Weak convergence theory for strong materials with p(x)-growth

Let L: Ω × R ^m × R ^{m × n} → R be a Caratheodory integrand with $c_1 |\nu |^{p(x)} + c_2 \leqslant L(x,u,\nu ) \leqslant c_3 |\nu |^{p(x)} + c_4 ,c_3 \geqslant c_1 > 0,n + \varepsilon \leqslant p( \cdot ) \leqslant p < \infty ,\varepsilon > 0.$ Under these assumptions the weak convergence theory holds for the integral functional $J(u): = \int\limits_\Omega {L(x,u(x),Du(x))dx} $ without further requirements. Weak convergence theory includes lower seraicontinuity with respect to the weak convergence of Sobolev functions, the convergence in energy property (weak convergence of Sobolev functions and convergence in energy imply the strong convergence of the functions), the integral representation for the relaxed energy and related questions. The results of the weak convergence theory follows from a characterization of gradient Young measures associated with these functionals.     

Lower semicontinuity and relaxation of signed functionals with linear growth in the context of {\mathcal A}-quasiconvexity

A lower semicontinuity and relaxation result with respect to weak-* convergence of measures is derived for functionals of the form $$\mu \in \mathcal{M}(\Omega; \mathbb{R}^d) \to \int \limits_\Omega f(\mu^a(x))\,{\rm {d}}x +\int \limits_\Omega f^\infty \left( \frac{{\rm{d}}\mu^s}{d|\mu^s|}(x)\right) \, d| \mu^s|(x),$$ where admissible sequences {μ _n } are such that ${\{{\mathcal{A}}\mu_{n}\}}$ converges to zero strongly in ${W^{-1 q}_{\rm loc}(\Omega)}$ and ${\mathcal {A}}$ is a partial differential operator with constant rank. The integrand f has linear growth and L ^∞-bounds from below are not assumed.     

(p,q)-odd digraphs

A digraph D is (p,q)-odd if and only if any subdivision of D contains a directed cycle of length different from p mod q. A characterization of (p,q)-odd digraphs analogous to the Seymour-Thomassen characterization of (1, 2)-odd digraphs is provided. In order to obtain this characterization we study the lattice generated by the directed cycles of a strongly connected digraph. We show that the sets of directed cycles obtained from an ear decomposition of the digraph in a natural way are bases of this lattice. A similar result does not hold for undirected graphs. However we construct, for each undirected 2-connected graph G, a set of cycles of G which form a basis of the lattice generated by the cycles of G. © 1996 John Wiley & Sons, Inc.     

Singular measures with smalH(p, q)-projections

We construct a singular probability measure μ on the complex sphere such that the Poisson integral of μ is a pluriharmonic function in the ball and the Fourier transform of μ is asp→∞. Supported by the Centre de Recerca Matemàtica (Institut d’Estudis Catalans, Barcelona) under a grant from DGICYT (Spain); partially supported by the RFFI grant 96-01-00693.     

Elementary aspects ofB p,q s (Kn) andF p,q s (Kn) spaces

In the first part of this paper, we discuss some properties of S^Ω(K_n), L _P ^Ω (K_n) and L _P ^Ω (K_n;l_q) spaces, give the Plancherel-Polya-Nikol’skij type inequalities and some multiplier theorems. In the second part of this paper, using the results of Part I we prove some preliminary results for the spaces B _p,q ^s (K_n) and F _p,q ^s (K_n).     

Degenerate Principal Series Representations of U(p, q) and Spin0(p, q)

Let p>q and let G be the group U(p, q) or Spin₀(p, q). Let P=LN be the maximal parabolic subgroup of G with Levi subgroup where

A lower semicontinuity result for some integral functionals in the space SBD

In this paper, we obtain a lower semicontinuity result with respect to the strong L1-convergence of the integral functionals F（u,Ω）=Ωf（x,u（x）,εu（x））dx defined in the space SBD of special functions with bounded deformation. Here Su represents the absolutely continuous part of the symmetrized distributional derivative Eu. The integrand f satisfies the standard growth assumptions of order p 〉 1 and some other conditions. Finally, by using this result,we discuss the existence of an constrained variational problem.