Addendum: Local Elliptic Regularity for the Dirichlet Fractional Laplacian

2017 ◽  
Vol 17 (4) ◽  
pp. 837-839 ◽  
Author(s):  
Umberto Biccari ◽  
Mahamadi Warma ◽  
Enrique Zuazua
Keyword(s):  
Dirichlet Problem ◽  
Sobolev Space ◽  
Besov Space ◽  
Weak Solutions ◽  
Fractional Laplacian ◽  
Local Regularity ◽  
Elliptic Regularity ◽  
Open Set ◽  
Regularity Of Weak Solutions

AbstractIn [1], for {1<p<\infty}, we proved the {W^{2s,p}_{\mathrm{loc}}} local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian {(-\Delta)^{s}} on an arbitrary bounded open set of {\mathbb{R}^{N}}. Here we make a more precise and rigorous statement. In fact, for {1<p<2} and {s\neq\frac{1}{2}}, local regularity does not hold in the Sobolev space {W^{2s,p}_{\mathrm{loc}}}, but rather in the larger Besov space {(B^{2s}_{p,2})_{\mathrm{loc}}}.

scholarly journals The Brezis–Nirenberg problem for nonlocal systems

2016 ◽  
Vol 5 (1) ◽  
Author(s):  
Luiz F. O. Faria ◽  
Olimpio H. Miyagaki ◽  
Fabio R. Pereira ◽  
Marco Squassina ◽  
Chengxiang Zhang
Keyword(s):  
Variational Methods ◽  
Weak Solutions ◽  
Fractional Laplacian ◽  
Critical Growth ◽  
Laplacian Operator ◽  
Nonlocal Equations ◽  
Suitable Sense ◽  
Regularity Of Weak Solutions ◽  
Fractional Laplacian Operator ◽  
Nirenberg Problem

AbstractBy means of variational methods we investigate existence, nonexistence as well as regularity of weak solutions for a system of nonlocal equations involving the fractional laplacian operator and with nonlinearity reaching the critical growth and interacting, in a suitable sense, with the spectrum of the operator.

scholarly journals Logarithmically Improved Regularity Criterion for the 3D Micropolar Fluid Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Hui Zhang
Keyword(s):  
Besov Space ◽  
Strong Solution ◽  
Weak Solutions ◽  
Micropolar Fluid ◽  
Regularity Criterion ◽  
Vorticity Field ◽  
Regularity Of Weak Solutions ◽  
Fluid Equations ◽  
Incompressible Micropolar Fluid ◽  
Micropolar Fluid Equations

We study the regularity of weak solutions to the incompressible micropolar fluid equations. We obtain an improved regularity criterion in terms of vorticity of velocity in Besov space. It is proved that if the vorticity field satisfies ∫0T∇×uB˙∞,∞0/1+log1+∇×uB˙∞,∞0dt<∞ then the strong solution can be smoothly extended after time T.

Generalized n-Laplacian: boundedness of weak solutions to the Dirichlet problem with nonlinearity in the critical growth range

2014 ◽  
Vol 12 (1) ◽  
Author(s):  
Robert Černý
Keyword(s):  
Dirichlet Problem ◽  
Continuous Function ◽  
Small Parameter ◽  
Orlicz Space ◽  
Weak Solutions ◽  
Critical Growth ◽  
Young Function ◽  
Bounded Set ◽  
Open Set ◽  
Continuous Potential

AbstractLet n ≥ 2 and let Ω ⊂ ℝn be an open set. We prove the boundedness of weak solutions to the problem $$u \in W_0^1 L^\Phi \left( \Omega \right) and - div\left( {\Phi '\left( {\left| {\nabla u} \right|} \right)\frac{{\nabla u}} {{\left| {\nabla u} \right|}}} \right) + V\left( x \right)\Phi '\left( {\left| u \right|} \right)\frac{u} {{\left| u \right|}} = f\left( {x,u} \right) + \mu h\left( x \right) in \Omega ,$$ where ϕ is a Young function such that the space W 01 L Φ(Ω) is embedded into an exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V(x) is a continuous potential, h ∈ L Φ(Ω) is a non-trivial continuous function and µ ≥ 0 is a small parameter. We consider two classical cases: the case of Ω being an open bounded set and the case of Ω = ℝn.

Infinitely many arbitrarily small positive solutions for the Dirichlet problem involving the p-Laplacian

2002 ◽  
Vol 132 (3) ◽  
pp. 511-519 ◽  
Author(s):  
Giovanni Anello ◽  
Giuseppe Cordaro
Keyword(s):  
Dirichlet Problem ◽  
Positive Solutions ◽  
Weak Solutions ◽  
Smooth Boundary ◽  
Open Set ◽  
Carathéodory Function ◽  
Image Position

In this paper we present a result of existence of infinitely many arbitrarily small positive solutions to the following Dirichlet problem involving the p-Laplacian, where Ω ∈ RN is a bounded open set with sufficiently smooth boundary ∂Ω, p > 1, λ > 0, and f: Ω × R → R is a Carathéodory function satisfying the following condition: there exists t̄ > 0 such that Precisely, our result ensures the existence of a sequence of a.e. positive weak solutions to the above problem, converging to zero in L∞(Ω).

Regularity of weak solutions under pure order differential operators by means of the continuous wavelet transform with rotations

2017 ◽  
Vol 15 (02) ◽  
pp. 1750010 ◽  
Author(s):  
Jaime Navarro ◽  
David Elizarraraz
Keyword(s):  
Wavelet Transform ◽  
Weak Solutions ◽  
Continuous Wavelet Transform ◽  
Differential Operators ◽  
Admissible Function ◽  
Partial Differential Operator ◽  
Rotation Parameter ◽  
Local Regularity ◽  
Continuous Wavelet ◽  
Regularity Of Weak Solutions

By introducing a rotation parameter besides translations and dilations, the continuous wavelet transform has been defined with respect to an admissible function not necessarily radial to study the local regularity of weak solutions [Formula: see text] under [Formula: see text], where [Formula: see text] is of class [Formula: see text] in a neighborhood of some point [Formula: see text] in [Formula: see text], and [Formula: see text] is a partial differential operator of order [Formula: see text]. The method used to study this fact is through the local convergence of the continuous wavelet transform.

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