A Strong Comparison Principle for the Dirichlet p-Laplacian

2020 ◽  
pp. 79-87 ◽  
Author(s):  
Mabel Cuesta ◽  
Peter Takáč
Keyword(s):  
Comparison Principle ◽  
Strong Comparison Principle

scholarly journals Strong Comparison Principle for the Fractional p-Laplacian and Applications to Starshaped Rings

2018 ◽  
Vol 18 (4) ◽  
pp. 691-704 ◽  
Author(s):  
Sven Jarohs
Keyword(s):  
Half Space ◽  
Comparison Principle ◽  
Nonnegative Function ◽  
Open Set ◽  
Nonnegative Solutions ◽  
Strong Comparison Principle

AbstractIn the following, we show the strong comparison principle for the fractional p-Laplacian, i.e. we analyze\quad\left\{\begin{aligned} \displaystyle(-\Delta)^{s}_{p}v+q(x)\lvert v\rvert% ^{p-2}v&\displaystyle\geq 0&&\displaystyle\phantom{}\text{in ${D}$},\\ \displaystyle(-\Delta)^{s}_{p}w+q(x)\lvert w\rvert^{p-2}w&\displaystyle\leq 0&% &\displaystyle\phantom{}\text{in ${D}$},\\ \displaystyle v&\displaystyle\geq w&&\displaystyle\phantom{}\text{in ${\mathbb% {R}^{N}}$},\end{aligned}\right.where {s\in(0,1)}, {p>1}, {D\subset\mathbb{R}^{N}} is an open set, and {q\in L^{\infty}(\mathbb{R}^{N})} is a nonnegative function. Under suitable conditions on s, p and some regularity assumptions on v, w, we show that either {v\equiv w} in {\mathbb{R}^{N}} or {v>w} in D. Moreover, we apply this result to analyze the geometry of nonnegative solutions in starshaped rings and in the half space.

The strong comparison principle in parabolic problems with thep-Laplacian in a domain

2019 ◽  
Vol 98 ◽  
pp. 365-373
Author(s):  
Jiří Benedikt ◽  
Petr Girg ◽  
Lukáš Kotrla ◽  
Peter Takáč
Keyword(s):  
Comparison Principle ◽  
Parabolic Problems ◽  
Strong Comparison Principle

scholarly journals Interior and boundary regularity results for strongly nonhomogeneous p,q-fractional problems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jacques Giacomoni ◽  
Deepak Kumar ◽  
Konijeti Sreenadh
Keyword(s):  
Maximum Principle ◽  
Comparison Principle ◽  
Weak Solutions ◽  
Global Regularity ◽  
Boundary Regularity ◽  
Holder Continuity ◽  
Regularity Of Weak Solutions ◽  
Regularity Results ◽  
Boundary Estimates ◽  
Strong Comparison Principle

Abstract In this article, we deal with the global regularity of weak solutions to a class of problems involving the fractional ( p , q ) {(p,q)} -Laplacian, denoted by ( - Δ ) p s 1 + ( - Δ ) q s 2 {(-\Delta)^{s_{1}}_{p}+(-\Delta)^{s_{2}}_{q}} for s 2 , s 1 ∈ ( 0 , 1 ) {s_{2},s_{1}\in(0,1)} and 1 < p , q < ∞ {1<p,q<\infty} . We establish completely new Hölder continuity results, up to the boundary, for the weak solutions to fractional ( p , q ) {(p,q)} -problems involving singular as well as regular nonlinearities. Moreover, as applications to boundary estimates, we establish a new Hopf-type maximum principle and a strong comparison principle in both situations.

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