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We establish pointwise bounds expressed in terms of a nonlinear potential of a generalized Wolff type for 풜 -superharmonic functions with nonlinear operator 풜 : Ω × ℝ n → ℝ n having measurable dependence on the spacial variable and Orlicz growth with respect to the last variable. The result is sharp as the same potential controls estimates from above and from below. Applying it we provide a bunch of precise regularity results including continuity and Hölder continuity for solutions to problems involving measures that satisfy conditions expressed in the natural scales. Finally, we give a variant of Hedberg–Wolff theorem on characterization of the dual of the Orlicz space. [ABSTRACT FROM AUTHOR]

We consider non-degenerate elliptic systems of the type -divA(x,Du)=g(x)in Ω⊂Rn, where u:Ω→RN, g∈L²(Ω,RN) and x→A(x,ξ) has derivatives in the Marcinkiewicz class Ln,∞(Ω) with sufficiently small distance to L∞(Ω). We prove that every weak solution u∈Wloc1,p​(Ω,RN) of the system is such that the nonlinear expression of its gradient Vμ​(Du):=(μ²+∣Du∣²)4/p-2​Du is weakly differentiable with D(Vμ​(Du))∈Lloc²​(Ω). Then, we deduce higher differentiability properties for u itself and some higher integrability results for its gradient. [ABSTRACT FROM AUTHOR]

We prove a lower bound on the number of the convex components of a compact set with non-empty interior in ℝ n for all n ≥ 2 . Our result generalizes and improves the inequalities previously obtained in [M. Carozza, F. Giannetti, F. Leonetti and A. Passarelli di Napoli, Convex components, Commun. Contemp. Math. 21 2019, 6, Article ID 1850036] and [M. La Civita and F. Leonetti, Convex components of a set and the measure of its boundary, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 56 2008/09, 71–78]. [ABSTRACT FROM AUTHOR]

We prove a lower bound on the number of the convex components of a compact set with non-empty interior in ℝ n for all n ≥ 2 . Our result generalizes and improves the inequalities previously obtained in [M. Carozza, F. Giannetti, F. Leonetti and A. Passarelli di Napoli, Convex components, Commun. Contemp. Math. 21 2019, 6, Article ID 1850036] and [M. La Civita and F. Leonetti, Convex components of a set and the measure of its boundary, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 56 2008/09, 71–78]. [ABSTRACT FROM AUTHOR]

We consider weak solutions $u:\Omega _T\to {\open R}^N$ to parabolic systems of the type $$u_t-{\rm div}\;a(x,t,Du) = 0\quad {\rm in}\;\Omega _T = \Omega \times (0,T),$$ where the function a (x , t , ξ) satisfies (p , q)-growth conditions. We give an a priori estimate for weak solutions in the case of possibly discontinuous coefficients. More precisely, the partial maps $x\mapsto a(x,t,\xi)$ under consideration may not be continuous, but may only possess a Sobolev-type regularity. In a certain sense, our assumption means that the weak derivatives $D_xa(\cdot ,\cdot ,\xi)$ are contained in the class $L^\alpha (0,T;L^\beta (\Omega))$ , where the integrability exponents $\alpha ,\beta $ are coupled by $$\displaystyle{{p(n + 2)-2n} \over {2\alpha }} + \displaystyle{n \over \beta } = 1-\kappa $$ for some κ ∈ (0,1). For the gap between the two growth exponents we assume $$2 \les p Under further assumptions on the integrability of the spatial gradient, we prove a result on higher differentiability in space as well as the existence of a weak time derivative $u_t\in L^{p/(q-1)}_{{\rm loc}}(\Omega _T)$. We use the corresponding a priori estimate to deduce the existence of solutions of Cauchy–Dirichlet problems with the mentioned higher differentiability property. [ABSTRACT FROM AUTHOR]