Abstract We prove that a certain discrete energy for triangulated surfaces, defined in the spirit of discrete differential geometry, converges to the Willmore energy in the sense of Γ-convergence. Variants of this discrete energy have been discussed before in the computer graphics literature.

Abstract Let 1 < p ≤ 4 {1<p\leq 4} when n = 1 {n=1} , and 1 < p < 3 + 1 n - 1 {1<p<3+\frac{1}{n-1}} when n ≥ 2 {n\geq 2} . We obtain the second-order horizontal Sobolev HW loc 2 , 2 {\operatorname{HW}^{2,2}_{\mathrm{loc}}} -regularity of p-harmonic functions in the Heisenberg group ℍ n {{\mathbb{H}}^{n}} . This improves the known range of p obtained by Domokos and Manfredi in 2005.

Abstract We study the fine properties of a class of weak solutionsuof the eikonal equation arising as asymptotic domain of a family of energy functionals introduced in [T. Rivière and S. Serfaty, Limiting domain wall energy for a problem related to micromagnetics, Comm. Pure Appl. Math. 54 2001, 3, 294–338]. In particular, we prove that the entropy defect measure associated touis concentrated on a 1-rectifiable set, which detects the jump-type discontinuities ofu.

Abstract We study quasilinear elliptic equations of the type - Δ p ⁢ u = σ ⁢ u q + μ {-\Delta_{p}u=\sigma u^{q}+\mu} in ℝ n {\mathbb{R}^{n}} in the case 0 < q < p - 1 {0<q<p-1} , where μ and σ are nonnegative measurable functions, or locally finite measures, and Δ p ⁢ u = div ⁡ ( | ∇ ⁡ u | p - 2 ⁢ ∇ ⁡ u ) {\Delta_{p}u=\operatorname{div}(\lvert\nabla u\rvert^{p-2}\nabla u)} is the p-Laplacian. Similar equations with more general local and nonlocal operators in place of Δ p {\Delta_{p}} are treated as well. We obtain existence criteria and global bilateral pointwise estimates for all positive solutions u: u ⁢ ( x ) ≈ ( 𝐖 p ⁢ σ ⁢ ( x ) ) p - q p - q - 1 + 𝐊 p , q ⁢ σ ⁢ ( x ) + 𝐖 p ⁢ μ ⁢ ( x ) , x ∈ ℝ n , u(x)\approx({\mathbf{W}}_{p}\sigma(x))^{\frac{p-q}{p-q-1}}+{\mathbf{K}}_{p,q}% \sigma(x)+{\mathbf{W}}_{p}\mu(x),\quad x\in\mathbb{R}^{n}, where 𝐖 p {{\mathbf{W}}_{p}} and 𝐊 p , q {{\mathbf{K}}_{p,q}} are, respectively, the Wolff potential and the intrinsic Wolff potential, with the constants of equivalence depending only on p, q, and n. The contributions of μ and σ in these pointwise estimates are totally separated, which is a new phenomenon even when p = 2 {p=2} .

Abstract Let Ω be a bounded domain in ℝ n {\mathbb{R}^{n}} . Under appropriate conditions on Ω, we prove existence and uniqueness of continuous functions solving the Dirichlet problem associated to certain nonlinear mean value properties in Ω with respect to balls of variable radius. We also show that, when properly normalized, such functions converge to the p-harmonic solution of the Dirichlet problem in Ω for p ⩾ 2 {p\geqslant 2} . Existence is obtained via iteration, a fundamental tool being the construction of explicit universal barriers in Ω.

Abstract Motivated by the vanishing contact problem, we study in the present paper the convergence of solutions of Hamilton–Jacobi equations depending nonlinearly on the unknown function. LetH⁢(x,p,u){H(x,p,u)}be a continuous Hamiltonian which is strictly increasing inu, and is convex and coercive inp. For each parameterλ>0{\lambda>0}, we denote byuλ{u^{\lambda}}the unique viscosity solution of the Hamilton–Jacobi equationH⁢(x,D⁢u⁢(x),λ⁢u⁢(x))=c.H\big{(}x,Du(x),\lambda u(x)\big{)}=c.Under quite general assumptions, we prove thatuλ{u^{\lambda}}converges uniformly, as λ tends to zero, to a specific solution of the critical Hamilton–Jacobi equationH⁢(x,D⁢u⁢(x),0)=c{H(x,Du(x),0)=c}. We also characterize the limit solution in terms of Peierls barrier and Mather measures.

Abstract Building upon the recent results in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] we provide a thorough description of the free boundary for the solutions to the fractional obstacle problem in ℝ n + 1 {\mathbb{R}^{n+1}} with obstacle function φ (suitably smooth and decaying fast at infinity) up to sets of null ℋ n - 1 {{\mathcal{H}}^{n-1}} measure. In particular, if φ is analytic, the problem reduces to the zero obstacle case dealt with in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] and therefore we retrieve the same results: (i) local finiteness of the ( n - 1 ) {(n-1)} -dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure), (ii) ℋ n - 1 {{\mathcal{H}}^{n-1}} -rectifiability of the free boundary, (iii) classification of the frequencies and of the blowups up to a set of Hausdorff dimension at most ( n - 2 ) {(n-2)} in the free boundary. Instead, if φ ∈ C k + 1 ⁢ ( ℝ n ) {\varphi\in C^{k+1}(\mathbb{R}^{n})} , k ≥ 2 {k\geq 2} , similar results hold only for distinguished subsets of points in the free boundary where the order of contact of the solution with the obstacle function φ is less than k + 1 {k+1} .

Abstract In this paper, we investigate the fine properties of functions under suitable geometric conditions on the jump set. Precisely, given an open set Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} and given p > 1 {p>1} , we study the blow-up of functions u ∈ GSBV ⁢ ( Ω ) {u\in\mathrm{GSBV}(\Omega)} , whose jump sets belong to an appropriate class 𝒥 p {\mathcal{J}_{p}} and whose approximate gradients are p-th power summable. In analogy with the theory of p-capacity in the context of Sobolev spaces, we prove that the blow-up of u converges up to a set of Hausdorff dimension less than or equal to n - p {n-p} . Moreover, we are able to prove the following result which in the case of W 1 , p ⁢ ( Ω ) {W^{1,p}(\Omega)} functions can be stated as follows: whenever u k {u_{k}} strongly converges to u, then, up to subsequences, u k {u_{k}} pointwise converges to u except on a set whose Hausdorff dimension is at most n - p {n-p} .

Abstract We consider a double phase problems with unbalanced growth and a superlinear reaction, which need not satisfy the Ambrosetti–Rabinowitz condition. Using variational tools and the Nehari method, we show that the Dirichlet problem has at least three nontrivial solutions, a positive solution, a negative solution and a nodal solution. The nodal solution has exactly two nodal domains.

Abstract In this paper, we consider the initial boundary value problem for a class of fourth-order wave equations with strong damping term, nonlinear weak damping term, strain term and nonlinear source term in polynomial form. First, the local solution is obtained by using fix point theory. Then, by constructing the potential well structure frame, we get the global existence, asymptotic behavior and blowup of solutions for the subcritical initial energy and critical initial energy respectively. Ultimately, we prove the blowup in finite time of solutions for the arbitrarily positive initial energy case.