arXiv subjects

Cristian Enache

Publications and source records attributed to Cristian Enache.

4 recordsLinked to original sources

arXiv2026

Minimum principles and a priori estimates for 2-Hessian problems

In this paper we investigate a class of $2$-Hessian equations and establish a minimum principle for a $P$-function in the sense of L.E. Payne (see R. Sperb \cite{Sp81}). The analysis is based on a sharp matrix inequality providing an estimate for a suitable combination of second-order partial derivatives of the solution. Exploiting this estimate, we derive a differential inequality for the associated $P$-function and obtain a minimum principle in higher dimensions under a convexity assumption. As an application of our results, together with convexity results established in X.-N. Ma and L. Xu \cite{MX08}, P. Liu, X.-N. Ma and L. Xu \cite{LMX10}, P. Salani \cite{Sa12}, and Y. Ye \cite{Ye13}, we derive a priori bounds for solutions of several classical $2$-Hessian boundary value problems.

math.AP↗

arXiv2026

Comparison results for the $p$-torsional rigidity on convex domains

For each open, bounded and convex domain $\Omega \subset \mathbb{R}^{D},$ $D\geq 2$, and each real number $p>1,$ we denote by $u_{p}$ the $p$\emph{-torsion function} on $\Omega $, i.e. the solution of the \emph{torsional creep problem} $\Delta_{p}u=-1$ in $\Omega $, $u=0$ on $\partial \Omega $, where $\Delta _{p}u:=\operatorname{div}( \left\vert \nabla u\right\vert ^{p-2}\nabla u) $ is the $p$-Laplacian. Let $T_p(\Omega)$ be the $p$\emph{-torsional rigidity} on $\Omega $, defined as $T_{p}\left( \Omega \right) :=\int_{\Omega }u_{p}dx$. Define $T\left( p;\Omega \right) :=\left\vert \Omega \right\vert ^{p-1}T_{p}\left( \Omega \right) ^{1-p}$, where $|\Omega|$ stands for the Lebesgue measure of $\Omega$. The main purpose of this paper is to compare the values of $T(p;\Omega)$ for bounded convex domains having different inradii. We prove that for any $0<a<b$ there exists a constant $\gamma_{D,p}\in[1/D,1)$, depending only on the dimension $D$ and the parameter $p$, such that $T(p;\Omega_b)\leq T(p;\Omega_a)$, for all $ \Omega_a\in\PP^D(a)$, and $\Omega_b\in\PP^D(b)$, if and only if $\gamma_{D,p}b\geq a$, where $\PP^D(r)$ denotes the family of convex bounded domains in $\mathbb{R}^D$ of inradius $r$. In addition, we discuss the asymptotic equality case, the limiting regimes $p\rightarrow 1^+$ and $p\rightarrow\infty$, and the sharpness of our bounds on model families such as rectangles, orthotopes, ellipses, and triangles}. We also derive a Saint-Venant type comparison result under additional geometric constraints, as a direct consequence of our main theorem.

math.AP↗

arXiv2026

Some minimum principles for a class of nonlinear elliptic problems in divergence form

In this paper we study a general class of nonlinear elliptic problems in divergence form. First, we prove that the solutions to these problems satisfy a convexity property when the given domain is strictly convex. Then, making use of this convexity property, we develop some minimum principles for an appropriate $P$-function, in the sense of L.~E.~Payne. Finally, this new minimum principle is applied to find a priori estimates for the solutions, in terms of the mean curvature of the boundary of the underlying domain.

math.AP↗

arXiv2019

Minimum principles and a priori estimates for some translating soliton type problems

In this paper we are dealing with two classes of mean curvature type problems that generalize the translating soliton problem. A first result proves that the solutions to these problems have unique interior critical points. Using this uniqueness result, we next derive a priori $C^0$ and $C^1$ estimates for the solutions to these problems, by means of some minimum principles for appropriate $P$-functions, in the sense of L.E. Payne.

math.DG↗