The geometric Neumann problem for the Liouville equation

Let Ω denote the upper half-plane ${\mathbb{R}_+^2}$ or the upper half-disk ${D_{\varepsilon}^+\subset \mathbb{R}_+^2}$ of center 0 and radius ${\varepsilon}$ . In this paper we classify the solutions ${v\in\;C^2(\overline{\Omega}\setminus\{0\})}$ to the Neumann problem $$\left\{\begin{array}{lll}{\Delta v+2 Ke^v=0\quad {\rm in}\,\Omega\subseteq \mathbb{R}^2_+=\{(s, t)\in \mathbb{R}^2: t >0 \},}\\ {\frac{\partial v}{\partial t}=c_1e^{v/2}\quad\quad\quad{\rm on}\,\partial\Omega\cap\{s >0 \},}\\ {\frac{\partial v}{\partial t}=c_2e^{v/2}\quad\quad\quad{\rm on}\,\partial\Omega\cap\{s <0 \},}\end{array}\right.$$ where ${K, c_1, c_2 \in \mathbb{R}}$ , with the finite energy condition ${\int_{\Omega} e^v < \infty}$ As a result, we classify the conformal Riemannian metrics of constant curvature and finite area on a half-plane that have a finite number of boundary singularities, not assumed a priori to be conical, and constant geodesic curvature along each boundary arc.     

Rellich inequalities with weights

Let Ω be a cone in ${\mathbb {R}^{n}}$ with n ≥? 2. For every fixed ${\alpha \in \mathbb {R}}$ we find the best constant in the Rellich inequality ${\int\nolimits_{\Omega}|x|^{\alpha}|\Delta u|^{2}dx \ge C\int\nolimits_{\Omega}|x|^{\alpha-4}|u|^{2}dx}$ for ${u \in C^{2}_{c}(\overline\Omega\setminus\{0\})}$ . We also estimate the best constant for the same inequality on ${C^{2}_{c}(\Omega)}$ . Moreover we show improved Rellich inequalities with remainder terms involving logarithmic weights on cone-like domains.     

Boundary partial regularity for a class of biharmonic maps

We consider the Dirichlet problem for biharmonic maps u from a bounded, smooth domain ${\Omega\subset\mathbb R^n (n\ge 5)}$ to a compact, smooth Riemannian manifold ${N\subset{\mathbb {R}}^l}$ without boundary. For any smooth boundary data, we show that if u is a stationary biharmonic map that satisfies a certain boundary monotonicity inequality, then there exists a closed subset ${\Sigma\subset\overline{\Omega}}$ , with ${H^{n-4}(\Sigma)=0}$ , such that ${\displaystyle u\in C^\infty(\overline\Omega\setminus\Sigma, N)}$ .     

Stability results for the N-dimensional Schiffer conjecture via a perturbation method

Given a eigenvalue $\mu _{0m}^2$ of $-\Delta $ in the unit ball $B_1$ , with Neumann boundary conditions, we prove that there exists a class $\mathcal{D}$ of $C^{0,1}$ -domains, depending on $\mu _{0m} $ , such that if $u$ is a no trivial solution to the following problem $ \Delta u+\mu u=0$ in $\Omega , u=0$ on $\partial \Omega $ , and $ \int \nolimits _{\partial \Omega }\partial _{\mathbf{n}}u=0$ , with $\Omega \in \mathcal{D}$ , and $\mu =\mu _{0m}^2+o(1)$ , then $\Omega $ is a ball. Here $\mu $ is a eigenvalue of $-\Delta $ in $\Omega $ , with Neumann boundary conditions.     

Existence and Multiplicity Results for Some Elliptic Systems in Unbounded Cylinders

We study the following nonlinear elliptic system of Lane–Emden type $$\left\{\begin{array}{ll} -\Delta u = {\rm sgn}(v) |v| ^{p-1} \qquad \qquad \qquad \; {\rm in} \; \Omega , \\ -\Delta v = - \lambda {\rm sgn} (u)|u| \frac{1}{p-1} + f(x, u)\; \; {\rm in}\; \Omega , \\ u = v = 0 \qquad \qquad \qquad \quad \quad \;\;\;\;\; {\rm on}\; \partial \Omega , \end{array}\right.$$ where ${\lambda \in \mathbb{R}}$ . If ${\lambda \geq 0}$ and ${\Omega}$ is an unbounded cylinder, i.e., ${\Omega = \tilde \Omega \times \mathbb{R}^{N-m} \subset \mathbb{R}^{N}}$ , ${N - m \geq 2, m \geq 1}$ , existence and multiplicity results are proved by means of the Principle of Symmetric Criticality and some compact imbeddings in partially spherically symmetric spaces. We are able to state existence and multiplicity results also if ${\lambda \in \mathbb{R}}$ and ${\Omega}$ is a bounded domain in ${\mathbb{R}^{N}, N \geq 3}$ . In particular, a good finite dimensional decomposition of the Banach space in which we work is given.     

On the boundary of the attainable set of the dirichlet spectrum

Denoting by ${\varepsilon\subseteq\mathbb{R}^2}$ the set of the pairs ${(\lambda_1(\Omega),\,\lambda_2(\Omega))}$ for all the open sets ${\Omega\subseteq\mathbb{R}^N}$ with unit measure, and by ${\Theta\subseteq\mathbb{R}^N}$ the union of two disjoint balls of half measure, we give an elementary proof of the fact that ${\partial\varepsilon}$ has horizontal tangent at its lowest point ${(\lambda_1(\Theta),\,\lambda_2(\Theta))}$ .     

On the singular set of minimizers of p(x)-energies

We treat the partial regularity of locally bounded local minimizers $u$ for the $p(x)$ -energy functional $$\begin{aligned} \mathcal{E }(v;\Omega ) = \int \left( g^{\alpha \beta }(x)h_{ij}(v) D_\alpha v^i (x) D_\beta v^j (x) \right) ^{p(x)/2} dx, \end{aligned}$$ defined for maps $v : \Omega (\subset \mathbb R ^m) \rightarrow \mathbb R ^n$ . Assuming the Lipschitz continuity of the exponent $p(x) \ge 2$ , we prove that $u \in C^{1,\alpha }(\Omega _0)$ for some $\alpha \in (0,1)$ and an open set $\Omega _0 \subset \Omega $ with $\dim _\mathcal{H }(\Omega \setminus \Omega _0) \le m-[\gamma _1]-1$ , where $\dim _\mathcal{H }$ stands for the Hausdorff dimension, $[\gamma _1]$ the integral part of $\gamma _1$ , and $\gamma _1 = \inf p(x)$ .     

Profile of the least energy solution of a singular perturbed Neumann problem with mixed powers

We consider the problem ${\varepsilon^{2}\Delta u - u^q + u^p = 0\,{\rm in}\,\Omega,\,u > 0\,{\rm in}\,\Omega,\,\frac{\partial u}{\partial \nu} = 0\,{\rm on}\,\partial\Omega }$ where Ω is a smooth bounded domain in ${\mathbb{R}^N}$ , ${1 < q < p < {N+2\over N-2}}$ if N ≥ 2 and ${\varepsilon}$ is a small positive parameter. We determine the location and shape of the least energy solution when ${\varepsilon \rightarrow 0.}$      

Uniqueness of integrable solutions to {\nabla \zeta=G \zeta, \zeta|_\Gamma = 0} for integrable tensor coefficients G and applications to elasticity

Let ${\Omega \subset \mathbb{R}^{N}}$ be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ${\partial\Omega}$ . We show that the solution to the linear first-order system $$\nabla \zeta = G\zeta, \, \, \zeta|_\Gamma = 0 \quad \quad \quad (1)$$ is unique if ${G \in \textsf{L}^{1}(\Omega; \mathbb{R}^{(N \times N) \times N})}$ and ${\zeta \in \textsf{W}^{1,1}(\Omega; \mathbb{R}^{N})}$ . As a consequence, we prove $$||| \cdot ||| : \textsf{C}_{o}^{\infty}(\Omega, \Gamma; \mathbb{R}^{3}) \rightarrow [0, \infty), \, \, u \mapsto \parallel {\rm sym}(\nabla uP^{-1})\parallel_{\textsf{L}^{2}(\Omega)}$$ to be a norm for ${P \in \textsf{L}^{\infty}(\Omega; \mathbb{R}^{3 \times 3})}$ with Curl ${P \in \textsf{L}^{p}(\Omega; \mathbb{R}^{3 \times 3})}$ , Curl ${P^{-1} \in \textsf{L}^{q}(\Omega; \mathbb{R}^{3 \times 3})}$ for some p, q > 1 with 1/p + 1/q = 1 as well as det ${P \geq c^+ > 0}$ . We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let ${\Phi \in \textsf{H}^{1}(\Omega; \mathbb{R}^{3})}$ satisfy sym ${(\nabla\Phi^\top\nabla\Psi) = 0}$ for some ${\Psi \in \textsf{W}^{1,\infty}(\Omega; \mathbb{R}^{3}) \cap \textsf{H}^{2}(\Omega; \mathbb{R}^{3})}$ with det ${\nabla\Psi \geq c^+ > 0}$ . Then, there exist a constant translation vector ${a \in \mathbb{R}^{3}}$ and a constant skew-symmetric matrix ${A \in \mathfrak{so}(3)}$ , such that ${\Phi = A\Psi + a}$ .     

Cohomology of Some Complex Laminations

In this paper we solve the ${\overline{\partial }}$ -problem along the leaves for two types of laminations: (i) Some open sets Ω of ${{\mathbb C}\times B}$ (where B is any differentiable manifold) endowed with the canonical foliation that is, the foliation whose leaves are the sections ${\Omega ^t=\{ z\in {\mathbb C}:(z,t)\in \Omega \}}$ . We construct a solution to the equation ${\overline{\partial }h=fd\overline z}$ for any function ${f:\Omega\longrightarrow {\mathbb C}}$ of class ${C^{s}\,(s\in \mathbb{N}\cup\{ \infty \}),\,C^\infty}$ along the leaves and satisfies some growth conditions near the singularities. (ii) A complex lamination by Riemann surfaces obtained by suspending a homeomorphism of a closed set of the Euclidean space ${\mathbb{C}\times \mathbb{R}}$ .     

Singular limits for Liouville-type equations on the flat two-torus

We consider the problem $$\begin{aligned} -\Delta u=\varepsilon ^{2}e^{u}- \frac{1}{|\Omega |}\int _\Omega \varepsilon ^{2} e^{u}+ {4\pi N\over |\Omega |} - 4 \pi N\delta _p, \quad \text{ in} {\Omega }, \quad \int _\Omega u=0 \end{aligned}$$ in a flat two-torus $\Omega $ with periodic boundary conditions, where $\varepsilon >0,\,|\Omega |$ is the area of the $\Omega $ , $N>0$ and $\delta _p$ is a Dirac mass at $p\in \Omega $ . We prove that if $1\le m<N+1$ then there exists a family of solutions $\{u_\varepsilon \}_{\varepsilon }$ such that $\varepsilon ^{2}e^{u_\varepsilon }\rightharpoonup 8\pi \sum _{i=1}^m\delta _{q_i}$ as $\varepsilon \rightarrow 0$ in measure sense for some different points $q_{1}, \ldots , q_{m}$ . Furthermore, points $q_i$ , $i=1,\dots ,m$ are different from $p$ .     

True Poly-Bergman and Poly-Bergman Kernels for the Complement of a Closed Disk

Let $k$ and $j$ be positive integers. We prove that the action of the two-dimensional singular integral operators $(S_\Omega )^{j-1}$ and $(S_\Omega ^*)^{j-1}$ on a Hilbert base for the Bergman space $\mathcal{A }^2(\Omega )$ and anti-Bergman space $\mathcal{A }^2_{-1}(\Omega ),$ respectively, gives Hilbert bases $\{ \psi _{\pm j , k } \}_{ k }$ for the true poly-Bergman spaces $\mathcal{A }_{(\pm j)}^2(\Omega ),$ where $S_\Omega $ denotes the compression of the Beurling transform to the Lebesgue space $L^2(\Omega , dA).$ The functions $\psi _{\pm j,k}$ will be explicitly represented in terms of the $(2,1)$ -hypergeometric polynomials as well as by formulas of Rodrigues type. We prove explicit representations for the true poly-Bergman kernels and more transparent representations for the poly-Bergman kernels of $\Omega $ . We establish Rodrigues type formulas for the poly-Bergman kernels of $\mathbb{D }$ .     

On additive involutions and Hamel bases

We provide an example of a discontinuous involutory additive function ${a: \mathbb{R}\to \mathbb{R}}$ such that ${a(H) \setminus H \ne \emptyset}$ for every Hamel basis ${H \subset \mathbb{R}}$ and show that, in fact, the set of all such functions is dense in the topological vector space of all additive functions from ${\mathbb{R}}$ to ${\mathbb{R}}$ with the Tychonoff topology induced by ${\mathbb{R}^{\mathbb{R}}}$ .     

A priori estimates for a class of degenerate elliptic equations

In this paper we investigate the regularity of solutions for the following degenerate partial differential equation $$\left \{\begin{array}{ll} -\Delta_p u + u = f \qquad {\rm in} \,\Omega,\\ \frac{\partial u}{\partial \nu} = 0 \qquad \qquad \,\,\,\,\,\,\,\,\,\, {\rm on} \,\partial \Omega, \end{array}\right.$$ when ${f \in L^q(\Omega), p > 2}$ and q ≥ 2. If u is a weak solution in ${W^{1, p}(\Omega)}$ , we obtain estimates for u in the Nikolskii space ${\mathcal{N}^{1+2/r,r}(\Omega)}$ , where r = q(p ? 2) + 2, in terms of the L ^q norm of f. In particular, due to imbedding theorems of Nikolskii spaces into Sobolev spaces, we conclude that ${\|u\|^r_{W^{1 + 2/r - \epsilon, r}(\Omega)} \leq C(\|f\|_{L^q(\Omega)}^q + \| f\|^{r}_{L^q(\Omega)} + \|f\|^{2r/p}_{L^q(\Omega)})}$ for every ${\epsilon > 0}$ sufficiently small. Moreover, we prove that the resolvent operator is continuous and compact in ${W^{1,r}(\Omega)}$ .     

Maximal, potential and singular operators in vanishing generalized Morrey spaces

We introduce vanishing generalized Morrey spaces ${V\mathcal{L}^{p,\varphi}_\Pi (\Omega), \Omega \subseteq \mathbb{R}^n}$ with a general function ${\varphi(x, r)}$ defining the Morrey-type norm. Here ${\Pi \subseteq \Omega}$ is an arbitrary subset in Ω including the extremal cases ${\Pi = \{x_0\}, x_0 \in \Omega}$ and Π = Ω, which allows to unify vanishing local and global Morrey spaces. In the spaces ${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n)}$ we prove the boundedness of a class of sublinear singular operators, which includes Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel. We also prove a Sobolev-Spanne type ${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n) \rightarrow V\mathcal{L}^{q,\varphi^\frac{q}{p}}_\Pi (\mathbb{R}^n)}$ -theorem for the potential operator I ^α . The conditions for the boundedness are given in terms of Zygmund-type integral inequalities on ${\varphi(x, r)}$ . No monotonicity type condition is imposed on ${\varphi(x, r)}$ . In case ${\varphi}$ has quasi- monotone properties, as a consequence of the main results, the conditions of the boundedness are also given in terms of the Matuszeska-Orlicz indices of the function ${\varphi}$ . The proofs are based on pointwise estimates of the modulars defining the vanishing spaces