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We propose interconnections between some problems of PDE, geometry, algebra, calculus and physics. Uniqueness of a solution of the Dirichlet problem and of some other boundary value problems for the string equation inside an arbitrary…
The present work is the first of a serie of two papers, in which we analyse the higher variational equations associated to natural Hamiltonian systems, in their attempt to give Galois obstruction to their integrability. We show that the…
For a bounded set $\Omega \subset \mathbb R^N$ and a perturbation $V \in C^1(\overline{\Omega})$, we analyze the concentration behavior of a blow-up sequence of positive solutions to \[ -\Delta u_\epsilon + \epsilon V = N(N-2)…
Maximization and minimization problems of the principle eigenvalue for divergence form second order elliptic operators with the Dirichlet boundary condition are considered. The principal eigen map of such elliptic operators is introduced…
We study the asymptotic Dirichlet problem for $\mathcal{A}$-harmonic functions on a Cartan-Hadamard manifold whose radial sectional curvatures outside a compact set satisfy an upper bound $$ K(P)\le - \frac{1+\varepsilon}{r(x)^2 \log r(x)}…
In this paper we study upper and lower bounds of the index and the nullity for sequences of harmonic maps with uniformly bounded Dirichlet energy from a two-dimensional Riemann surface into a compact target manifold. The main difficulty…
We study the Cauchy-Dirichlet pbm for superquadratic viscous Hamilton-Jacobi eq. We give a complete classification, namely rates and space-time profiles, in 1d case when viscosity sol. undergo gradient blow-up (GBU) or recovery of boundary…
We prove the multiplicity of spike-layer type solutions to the Dirichlet problem for the equation $-\Delta_p u = u^{q-1}$ in expanding annuli.
We study time-periodic solutions for the cubic wave equation on an interval with Dirichlet boundary conditions. We begin by following the perturbative construction of Vernov and Khrustalev and provide a rigorous derivation of the…
Parabolic equations with homogeneous Dirichlet conditions on the boundary are studied in a setting where the solutions are required to have a prescribed change of the profile in fixed time, instead of a Cauchy condition. It is shown that…
We present a numerical approximation method for linear diffusion-reaction problems with possibly discontinuous Dirichlet boundary conditions. The solution of such problems can be represented as a linear combination of explicitly known…
We analyze the set of real and complex Hadamard matrices with additional symmetry constrains. In particular, we link the problem of existence of maximally entangled multipartite states of $2k$ subsystems with $d$ levels each to the set of…
We introduce and study the Dirichlet problem for double divergence form elliptic equations with coefficients of low regularity and boundary conditions given by general Borel measures. Under broad assumptions we establish the solvability of…
In this PhD thesis, we give a new geometric approach to higher Teichm\"uller theory. In particular we construct a geometric structure on surfaces, generalizing the complex structure, and we explore its link to Hitchin components. The…
In this paper we study the construction of a discrete solution for a hyperbolic system of partial differentials of the strongly coupled type. In its construction, the discrete separation of matricial variable method was followed. Two…
We establish gradient estimates for solutions to the Dirichlet problem for the constant mean curvature equation in hyperbolic space. We obtain these estimates on bounded strictly convex domains by using the maximum principles theory of…
We consider the harmonic map heat flow for maps from the plane to the two-sphere. It is known that solutions to the initial value problem exhibit bubbling along a well-chosen sequence of times. We prove that every sequence of times admits a…
This paper presents the design and analysis of a Hybrid High-Order (HHO) approximation for a distributed optimal control problem governed by the Poisson equation. We propose three distinct schemes to address unconstrained control problems…
Using variational methods, we establish existence of multi-bump solutions for the following class of problems $$ \left\{ \begin{array}{l} \Delta^2 u +(\lambda V(x)+1)u = f(u), \quad \mbox{in} \quad \mathbb{R}^{N}, u \in…
In this article, we improve the convergence order of some finite volume solutions approximating some second order elliptic problems. We prove that finite volume approximations of order $O(h^{k+1})$, with $k$ integer, can be obtained after…