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Register a new userPrincipal eigenvalue and maximum principle for cooperative periodic-parabolic systems
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This paper classifies the set of supersolutions of a general class of periodic-parabolic problems in the presence of a positive supersolution. From this result we characterize the positivity of the underlying resolvent operator through the positivity of the associated principal eigenvalue and the existence of a positive strict supersolution. Lastly, this (scalar) characterization is used to characterize the strong maximum principle for a class of periodic-parabolic systems of cooperative type under arbitrary boundary conditions of mixed type.
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We consider general linear non-degenerate weakly-coupled cooperative elliptic systems and study certain monotonicity properties of the generalized principal eigenvalue in $mathbb{R}^d$ with respect to the potential. It is shown that monotonicity on the right is equivalent to the recurrence property of the twisted operator which is, in turn, equivalent to the minimal growth property at infinity of the principal eigenfunctions. The strict monotonicity property of the principal eigenvalue is shown to be equivalent with the exponential stability of the twisted operators. An equivalence between the monotonicity property on the right and the stochastic representation of the principal eigenfunction is also established.
In this article we find necessary and sufficient conditions for the strong maximum principle and compact support principle for non-negative solutions to the quasilinear elliptic inequalities $$Delta_infty u + G(|Du|) - f(u),leq 0quad text{in}; mathcal{O},$$ and $$Delta_infty u + G(|Du|) - f(u),geq 0quad text{in}; mathcal{O},$$ where $mathcal{O}$ denotes the infinity Laplacian, $G$ is an appropriate continuous function and $f$ is a nondecreasing, continuous function with $f(0)=0$.
We investigate the parabolic Boundary Harnack Principle for both divergence and non-divergence type operators by the analytical methods we developed in the elliptic context. Besides the classical case, we deal with less regular space-time domains, including slit domains.
In this paper we study the following parabolic system begin{equation*} Delta u -partial_t u =|u|^{q-1}u,chi_{{ |u|>0 }}, qquad u = (u^1, cdots , u^m) , end{equation*} with free boundary $partial {|u | >0}$. For $0leq q<1$, we prove optimal growth rate for solutions $u $ to the above system near free boundary points, and show that in a uniform neighbourhood of any a priori well-behaved free boundary point the free boundary is $C^{1, alpha}$ in space directions and half-Lipschitz in the time direction.
A shape optimization problem arising from the optimal reinforcement of a membrane by means of one-dimensional stiffeners or from the fastest cooling of a two-dimensional object by means of ``conducting wires is considered. The criterion we consider is the maximization of the first eigenvalue and the admissible classes of choices are the one of one-dimensional sets with prescribed total length, or the one where the constraint of being connected (or with an a priori bounded number of connected components) is added. The corresponding relaxed problems and the related existence results are described.
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