Partial regularity for minimisers of certain functionals having nonquadratic growth

We prove partial regularity results for local minimisers of

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

Everywhere regularity for a class of vectorial functionals under subquadratic general growth conditions

We consider the integral functional of the calculus of variations

     

Partial regularity for polyconvex functionals depending on the Hessian determinant

We prove a C ^2,α partial regularity result for local minimizers of polyconvex variational integrals of the type , where Ω is a bounded open subset of , and is a convex function, with subquadratic growth.     

Quasiminima of degenerate functionals with non polynomial growth

We prove a regularity theorem for quasiminima of a degenerate functional of the type , whereA (t) has non polynomial growth andb(x) is a weight belonging to theA, class of Muckenhoupt. Si dimostra un teorema di regolarità per i quasiminimi di un funzionale degenere del tipo , conA(t) ad andamento non polinomiale eb(x) peos della classeA, di Muckenhoupt.

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(Conferenza tenuta il 13 novembre 1989)     

Partial regularity for parabolic systems with non-standard growth

We study non-linear parabolic systems with non-standard p(z)-growth conditions and establish that the gradient of weak solutions is locally H?lder continuous with H?lder exponent b ? (0,1){\beta \in (0,1)} with respect to the parabolic metric on an open set of full Lebesgue measure, provided the exponent function p(z) itself is H?lder continuous with exponent β with respect to the parabolic metric.     

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.     

Partial regularity for certain classes of polyconvex functionals related to nonlinear elasticity

Consider the variational integral where Ω⊂ℝ ⁿ andp≥n≥2. H: (0, ∞)→[0, ∞) is a smooth convex function such that . We approximateJ by a sequence of regularized functionalsJ _δ whose minimizers converge strongly to anJ-minimizing function and prove partial regularity results forJ _δ-minimizers.     

Maximal regularity for non‐autonomous Robin boundary conditions

We consider a non‐autonomous Cauchy problem where is associated with the form , where V and H are Hilbert spaces such that V is continuously and densely embedded in H. We prove H‐maximal regularity, i.e., the weak solution u is actually in (if and ) under a new regularity condition on the form with respect to time; namely Hölder continuity with values in an interpolation space. This result is best suited to treat Robin boundary conditions. The maximal regularity allows one to use fixed point arguments to some non linear parabolic problems with Robin boundary conditions.     

Partial regularity for nonlinear elliptic systems with continuous growth exponent

For weak solutions of nonlinear elliptic systems of the type ${- {\rm div}a(x, u(x), Du(x)) = 0,}$ with nonstandard p(x) growth, we show interior partial Hölder continuity for any Hölder exponent ${\alpha \in (0,1)}$ , provided that the exponent function is ‘logarithmic Hölder continuous’. The result also covers the up to now open partial regularity for systems with constant growth with exponent p less than two in the case of merely continuous dependence on the spacial variable x.     

Partial regularity results for evolutional p-Laplacian systems with natural growth

 We study a regularity for evolutional p-Laplacian systems with natural growth on the gradient. It is shown that weak solutions of small image and their gradients are partial H?lder continuous and the size of the exceptional set for regularity is estimated in terms of Hausdorff measure. The main ingredient is to improve the Gehring inequality, which implies the higher integrability of the gradient and was first developed by Kinnunen and Lewis, so as to be well-worked in our perturbation estimate. We also use a refinement of the perturbation argument and make a device for H?lder estimates of the gradient. Received: 4 March 2002 Mathematics Subject Classification (2000): Primary 35D10, 35B65, 35K65     

Harnack inequality for minimizers of integral functionals with general growth conditions

We prove the Harnack inequality for positive minimizers of a class of integral functionals with non-standard growth conditions.     

Sharp regularity for functionals with (p,q) growth

We prove regularity theorems for minimizers of integral functionals of the Calculus of Variations

     

Partial regularity of minimizers of quasiconvex integrals with subquadratic growth

We prove partial regularity for minimizers of quasiconvex integrals of the form dx where the integral F() has subquadratic growth, ie .Research supported by MURST, Gruppo Nazionale 40%.     

The partial regularity for minimizers of splitting type variational integrals under general growth conditions i. the autonomous case

We consider splitting type variational problems with general growth conditions and prove the partial regularity (and the full regularity in 2D) of minimizers in the case of x-dependence. The results obtained generalize the results of Bildhauer and Fuchs concerning such problems with power growth conditions. Bibliography: 17 titles.     

Onsager Machlup functionals for non trace class SPDE's

Summary An Onsager Machlup functional limit is derived for a class of SPDE's whose principal part is not trace class. Both nondegenerate and degenerate limits are obtained, and are illustrated by examples. The proof uses FKG type inequalities.The work of this author was partially supported by the Bernstein Fund for the promotion of research at the TechnionThe work of this author was partially supported by the Center for Intelligent Control Systems at MIT under US Army research office grant DAAL03-86-K0171