Partial regularity for minimizers of discontinuous quasiconvex integrals with general growth

Regularity of Relaxed Minimizers of Quasiconvex Variational Integrals with (p,q)-Growth

We consider autonomous integrals F [u] := ∫ Ω f (Du)dx for u : R ⊃ Ω →R in the multidimensional calculus of variations, where the integrand f is a strictly quasiconvex C2-function satisfying the (p,q)-growth conditions γ|A| ≤ f (A)≤ Γ (1+ |A|) for every A ∈R with exponents 1 < p ≤ q < ∞. We examine the Lebesgue-Serrin extension Floc[u] := inf { liminf k→∞ F[uk] : W 1,q loc ∋ uk −−−⇀ k→∞ u weakl...

A simple partial regularity proof for minimizers of variational integrals

We consider multi-dimensional variational integrals F [u] := Ω f (·, u, Du) dx where the integrand f is a strictly convex function of its last argument. We give an elementary proof for the partial C 1,α-regularity of minimizers of F. Our approach is based on the method of A-harmonic approximation, avoids the use of Gehring's lemma, and establishes partial regularity with the optimal Hölder expo...

Partial Regularity for Almost Minimizers of Quasi-Convex Integrals

We consider almost minimizers of variational integrals whose integrands are quasiconvex. Under suitable growth conditions on the integrand and on the function determining the almost minimality, we establish almost everywhere regularity for almost minimizers and obtain results on the regularity of the gradient away from the singular set. We give examples of problems from the calculus of variatio...

Partial Regularity for Minimizers of Quasi-convex Functionals with General Growth

holds for every A ∈ R and every smooth ξ : B1 → R with compact support in the open unit ball B1 in R . By Jensen’s inequality, quasi convexity is a generalization of convexity. It was originally introduced as a notion for proving the lower semicontinuity and the existence of minimizers of variational integrals. In fact, assuming a power growth condition, quasi convexity is proved to be a necess...

Partial regularity for quasi minimizers