Potential estimates for elliptic systems with subquadratic growth

Measure data systems with Orlicz growth

“…By (73), (74) and Vitali's convergence theorem, we infer that the left-hand side of (33) converges to the left-hand side of (11). The right-hand side of (33) converges as in (32) to the right-hand side of (33).…”

“…Step 7. Limit function u u u is an approximable solution of (11). From (48), via Fatou's lemma, we deduce that Ω g(|Du u u|) dx ≤ c|µ µ µ|(Ω).…”

“…By (29) W 1,G (Ω, R m ) ⊂ L ∞ (Ω, R m ). Since we know that (11) holds true, to prove that an approximable solution u u u is a weak solution it is enough to show that it belongs to W 1,G (Ω, R m ). From (42) we directly deduce that u u u j L ∞ (Ω,R m ) ≤ t 1…”

“…See also [25,[43][44][45][46][47]59]. Simultaneously, the theory of regularity to such problems was being developed [4,11,22,24,30,31,41,42,51] as well as to related vectorial problems in calculus of variation [6,17,26,27,29]. For broader overview see [13,40,50].…”

Quasiconvexity and partial regularity via nonlinear potentials

“…As far as we know, this is the first paper where such an approach is taken in the quasiconvex case. Partial regularity results of this kind for elliptic systems have been derived by Kuusi & Mingione in [39] and Byun & Youn in [7]. Although our results are already completely new in the case of quasiconvex functionals with ppolynomial growth, i.e., F (z) ≈ |z| p , we shall treat the most general case of nonuniformly elliptic functionals with different polynomial growth conditions, that is…”