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Register a new userHopfs lemma for viscosity solutions to a class of non-local equations with applications
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We consider a large family of integro-differential equations and establish a non-local counterpart of Hopfs lemma, directly expressed in terms of the symbol of the operator. As closely related problems, we also obtain a variety of maximum principles for viscosity solutions. In our approach we combine direct analysis with functional integration, allowing a robust control around the boundary of the domain, and make use of the related ascending ladder height-processes. We then apply these results to a study of principal eigenvalue problems, the radial symmetry of the positive solutions, and the overdetermined non-local torsion equation.
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We provide sufficient conditions on the coefficients of a stochastic evolution equation on a Hilbert space of functions driven by a cylindrical Wiener process ensuring that its mild solution is positive if the initial datum is positive. As an application, we discuss the positivity of forward rates in the Heath-Jarrow-Morton model via Musielas stochastic PDE.
In this paper, we consider the existence and asymptotic properties of solutions to the following Kirchhoff equation begin{equation}label{1} onumber - Bigl(a+bint_{{R^3}} {{{left| { abla u} right|}^2}}Bigl) Delta u =lambda u+ {| u |^{p - 2}}u+mu {| u |^{q - 2}}u text { in } mathbb{R}^{3} end{equation} under the normalized constraint $int_{{mathbb{R}^3}} {{u}^2}=c^2$, where $a!>!0$, $b!>!0$, $c!>!0$, $2!<!q!<!frac{14}{3}!<! p!leq!6$ or $frac{14}{3}!<!q!<! p!leq! 6$, $mu!>!0$ and $lambda!in!R$ appears as a Lagrange multiplier. In both cases for the range of $p$ and $q$, the Sobolev critical exponent $p!=!6$ is involved and the corresponding energy functional is unbounded from below on $S_c=Big{ u in H^{1}({mathbb{R}^3}): int_{{mathbb{R}^3}} {{u}^2}=c^2 Big}$. If $2!<!q!<!frac{10}{3}$ and $frac{14}{3}!<! p!<!6$, we obtain a multiplicity result to the equation. If $2!<!q!<!frac{10}{3}!<! p!=!6$ or $frac{14}{3}!<!q!<! p!leq! 6$, we get a ground state solution to the equation. Furthermore, we derive several asymptotic results on the obtained normalized solutions. Our results extend the results of N. Soave (J. Differential Equations 2020 $&$ J. Funct. Anal. 2020), which studied the nonlinear Schr{o}dinger equations with combined nonlinearities, to the Kirchhoff equations. To deal with the special difficulties created by the nonlocal term $({int_{{R^3}} {left| { abla u} right|} ^2}) Delta u$ appearing in Kirchhoff type equations, we develop a perturbed Pohozaev constraint approach and we find a way to get a clear picture of the profile of the fiber map via careful analysis. In the meantime, we need some subtle energy estimates under the $L^2$-constraint to recover compactness in the Sobolev critical case.
We prove unique continuation properties of solutions to a large class of nonlinear, non-local dispersive equations. The goal is to show that if $u_1,,u_2$ are two suitable solutions of the equation defined in $mathbb R^ntimes[0,T]$ such that for some non-empty open set $Omegasubset mathbb R^ntimes[0,T]$, $u_1(x,t)=u_2(x,t)$ for $(x,t) in Omega$, then $u_1(x,t)=u_2(x,t)$ for any $(x,t)inmathbb R^ntimes[0,T]$. The proof is based on static arguments. More precisely, the main ingredient in the proofs will be the unique continuation properties for fractional powers of the Laplacian established by Ghosh, Salo and Ulhmann in cite{GhSaUh}, and some extensions obtained here.
This paper introduces a convenient solution space for the uniformly elliptic fully nonlinear path dependent PDEs. It provides a wellposedness result under standard Lipschitz-type assumptions on the nonlinearity and an additional assumption formulated on some partial differential equation defined locally by freezing the path.
We shall establish the interior Holder continuity for locally bounded weak solutions to a class of parabolic singular equations whose prototypes are begin{equation} u_t= abla cdot bigg( | abla u|^{p-2} abla u bigg), quad text{ for } quad 1<p<2, end{equation} and begin{equation} u_{t}- abla cdot ( u^{m-1} | abla u |^{p-2} abla u ) =0 , quad text{for} quad m+p>3-frac{p}{N}, end{equation} via a new and simplified proof using recent techniques on expansion of positivity and $L^{1}$-Harnack estimates.
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