Band 13, Heft 1

We prove the local regularity of solutions to the problem of minimizing ∫Ω[L∞⁢(∇⁡v⁢(x))+f⁢(x)⁢v⁢(x)]⁢𝑑x on ⁢u0+W01,∞⁢(Ω),\int_{\Omega}\bigl{[}L_{\infty}(\nabla v(x))+f(x)v(x)\bigr{]}\,dx\quad\text{on% }u^{0}+W^{1,\infty}_{0}(\Omega), where L∞{L_{\infty}} is either 12⁢|ξ|2{\frac{1}{2}\lvert\xi\rvert^{2}} for |ξ|≤1{\lvert\xi\rvert\leq 1} and +∞{+\infty} for |ξ|>1{\lvert\xi\rvert>1}, or a more general convex, extended-valued function.

In this paper we prove symmetry results for minimizers of a noncoercive functional defined on the class of Sobolev functions with zero mean value. We prove that the minimizers are foliated Schwarz symmetric, i.e., they are axially symmetric with respect to an axis passing through the origin and nonincreasing in the polar angle from this axis. In the two-dimensional case, we show a symmetry breaking.

The small-deformation limit of finite elasticity is considered in presence of a given crack. The rescaled finite energies with the constraint of global injectivity are shown to Γ-converge to the linearized elastic energy with a local constraint of non-interpenetration along the crack.

We obtain necessary and sufficient conditions for the existence of a positive finite energy solution to the inhomogeneous quasilinear elliptic equation -Δp⁢u=σ⁢uq+μ on ⁢ℝn-\Delta_{p}u=\sigma u^{q}+\mu\quad\text{on }\mathbb{R}^{n} in the sub-natural growth case 0<q<p-1{0<q<p-1}, where Δp{\Delta_{p}} (1<p<∞{1<p<\infty}) is the p -Laplacian, and σ, μ are positive Borel measures on ℝn{\mathbb{R}^{n}}. Uniqueness of such a solution is established as well. Similar inhomogeneous problems in the sublinear case 0<q<1{0<q<1} are treated for the fractional Laplace operator (-Δ)α{(-\Delta)^{\alpha}} in place of -Δp{-\Delta_{p}}, on ℝn{\mathbb{R}^{n}} for 0<α<n2{0<\alpha<\frac{n}{2}}, and on an arbitrary domain Ω⊂ℝn{\Omega\subset\mathbb{R}^{n}} with positive Green’s function in the classical case α=1{\alpha=1}.

The aim of the present paper is to study the boundedness of different conical square functions that arise naturally from second-order divergence form degenerate elliptic operators. More precisely, let Lw=-w-1⁢div(w⁢A⁢∇){L_{w}=-w^{-1}\mathop{\rm div}(wA\nabla)}, where w∈A2{w\in A_{2}} and A is an n×n{n\times n} bounded, complex-valued, uniformly elliptic matrix. Cruz-Uribe and Rios solved the L2⁢(w){L^{2}(w)}-Kato square root problem obtaining that Lw{\sqrt{L_{w}}} is equivalent to the gradient on L2⁢(w){L^{2}(w)}. The same authors in collaboration with the second named author of this paper studied the Lp⁢(w){L^{p}(w)}-boundedness of operators that are naturally associated with Lw{L_{w}}, such as the functional calculus, Riesz transforms, and vertical square functions. The theory developed admitted also weighted estimates (i.e., estimates in Lp⁢(v⁢d⁢w){L^{p}(v\,dw)} for v∈A∞⁢(w){v\in A_{\infty}(w)}), and in particular a class of “degeneracy” weights w was found in such a way that the classical L2{L^{2}}-Kato problem can be solved. In this paper, continuing this line of research, and also that originated in some recent results by the second and third named authors of the current paper, we study the boundedness on Lp⁢(w){L^{p}(w)} and on Lp⁢(v⁢d⁢w){L^{p}(v\,dw)}, with v∈A∞⁢(w){v\in A_{\infty}(w)}, of the conical square functions that one can construct using the heat or Poisson semigroup associated with Lw{L_{w}}. As a consequence of our methods, we find a class of degeneracy weights w for which L2{L^{2}}-estimates for these conical square functions hold. This opens the door to the study of weighted and unweighted Hardy spaces and of boundary value problems associated with Lw{L_{w}}.