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Given a bounded Lipschitz domain $\omega\subset\mathbb{R}^{d-1}$ and a lower semicontinuous function $W:\mathbb{R}^N\to\mathbb{R}_+\cup\{+\infty\}$ that vanishes on a finite set and that is bounded from below by a positive constant at…

Recent work in the literature has studied fourth-order elliptic operators on manifolds with boundary. This paper proves that, in the case of the squared Laplace operator, the boundary conditions which require that the eigenfunctions and…

Open effective field theories provide a systematic framework for describing physical systems interacting with an environment whose microscopic details are unknown, unobservable, or uncalculable. A basic step in constructing any effective…

In this work it is studied a quasilinear elliptic problem in the whole space $\mathbb{R}^N$ involving the $1-$Laplacian operator, with potentials which can vanish at infinity. The Euler-Lagrange functional is defined in a space whose…

If $(M,g)$ is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes of order $o(\lambda)$ saturating sup-norm estimates. In particular, it gives optimal conditions…

We prove sufficient and necessary conditions ensuring zero duality gap for Lagrangian duality in some classes of nonconvex optimization problems. To this aim, we use the $\Phi$-convexity theory and minimax theorems for $\Phi$-convex…

Adapting the method of Andrews-Clutterbuck we prove an eigenvalue gap theorem for a class of non symmetric second order linear elliptic operators on a convex domain in euclidean space. The class of operators includes the Bakry-Emery…

In the contest of optimal control problems, regularity results for optima are known when addressing fiber-strictly convex Lagrangian. For infinite time horizons, or for settings with infinite dimensional dynamics, the equivalence between…

It is well known that, the existence of a Lyapunov function is a sufficient condition for stability, asymptotic stability, or global asymptotic stability of an equilibrium point of an autonomous system $\dot{\mathbf{x}} = f(\mathbf{x})$. In…

We establish an explicit maximum principle for the Dirichlet problem associated with the $p$-Laplacian ($p>1$), where the constant depends on both $p$ and the geometry of the domain. From this result we derive two main applications. First,…

In nonlinear electrodynamics, by implementing the causality principle as the requirement that the group velocity of elementary excitations over a background field should not exceed the speed of light in the vacuum and the unitarity…

For the fractional Laplace equation, a surprising observation is the non-uniqueness for the basic Dirichlet type problems. In this paper, a somewhat sharp uniqueness condition for the fractional Laplace equation is established. We derive…

It is investigated the existence of a separately continuous function $f:X\times Y\to \mathbb R$ with an onepoint set of discontinuity for topological spaces $X$ and $Y$ which satisfy compactness type conditions. In particular, it is shown…

We study sharp conditions for the existence and nonexistence of infinitely many nonnegative solutions to the problem $-\Delta_p u = \lambda f(u)$ in a bounded domain with Dirichlet boundary conditions, where $f$ is a continuous function…

We consider the Dirichlet Laplacian for a strip in $\,\R^2$ with one straight boundary and a width $\,a(1+\lambda f(x))\,$, where $\,f\,$ is a smooth function of a compact support with a length $\,2b\,$. We show that in the critical case,…

Consider the Laplacian in a bounded domain in R^d with general (mixed) homogeneous boundary conditions. We prove that its eigenfunctions are `quasi-orthogonal' on the boundary with respect to a certain norm. Boundary orthogonality is proved…

In this paper, we propose necessary and sufficient conditions for a scalar function to be nonincreasing along solutions to general differential inclusions with state constraints. The problem of determining if a function is nonincreasing…

For a finite dimensional algebra $A$ with $0 < \phi dim (A) = m < \infty$ we prove that there always exist modules $M$ and $N$ such that $\phi(M) = m-1$ and $\phi (N) = 1$. On the other hand, we see an example of an algebra that not every…

Conditions for the existence of at least three positive solutions to the nonlinear first-order problem with a nonlinear nonlocal boundary condition given by && y'(t) - p(t)y(t) = \sum_{i=1}^m f_i\big(t,y(t)\big), \quad t\in[0,1], && \lambda…

We construct, for a homogeneous Lagrangian of arbitrary order in two independent variables, a differential 2-form with the property that it is closed precisely when the Lagrangian is null. This is similar to the property of the `fundamental…