In this paper, we study the existence of distributional solutions of the following non-local elliptic problem \begin{eqnarray*}
\left\lbrace
\begin{array}{l}
(-Δ)^{s}u + |\nabla u|^{p} =f \quad\text{ in } Ω
\qquad \qquad \qquad \,\,\, u=0 \,\,\,\,\,\,\,\text{ in } \mathbb{R}^{N}\setminus Ω, \quad s \in (1/2, 1).
\end{array}
\right. \end{eqnarray*}
We are interested in the relation between the regularity of the source term $f$, and the regularity of the corresponding solution. If $p<2s$, that is the natural growth, we are able to show the existence for all $f\in L^1(Ø)$. In the subcritical case, that is, for $p < p_{*}:=N/(N-2s+1)$, we show that solutions are $\mathcal{C}^{1, α}$ for $f \in L^{m}$, with $m$ large enough. In the general case, we achieve the same result under a condition on the size of the source. As an application, we may show that for regular sources, distributional solutions are viscosity solutions, and conversely.
It is established $L^{p}$ estimates for the fractional $Φ$-Laplacian operator defined in bounded domains where the nonlinearity is subcritical or critical in a suitable sense. Furthermore, using some fine estimates together with the Moser's iteration, we prove that any weak solution for fractional $Φ$-Laplacian operator defined in bounded domains belongs to $L^\infty(Ω)$ under appropriate hypotheses on the $N$-function $Φ$. Using the Orlicz space and taking into account the fractional setting for our problem the main results are stated for a huge class of nonlinear operators and nonlinearities…
Optimal estimates on asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to the following quasilinear elliptic equations \begin{eqnarray*} -Δ_{p}u-\fracμ{|x|^{p}}|u|^{p-2}u+m|u|^{p-2}u=f(u), & & x\in\R^{N}, \end{eqnarray*} where $1<p<N$, $0\leqμ<\left((N-p)/p\right)^{p}$, $m>0$ and $f$ is a continuous function.
Optimal estimates on asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to the following quasilinear elliptic equations \begin{eqnarray*} -Δ_{p}u-\fracμ{|x|^{p}}|u|^{p-2}u+m|u|^{p-2}u=f(u), & & x\in\R^{N}, \end{eqnarray*} where $1<p<N,0\leqμ<\left((N-p)/p\right)^{p}$, $m>0$ and $f$ is a nonlinear function.
In this paper, the aim of our work is to establish global weighted gradient estimates via fractional maximal functions and the point-wise regularity estimates of Dirichlet problem for divergence elliptic equations of the type \begin{align*} \mathrm{div}(A(x,\nabla u)) = \mathrm{div}(f) \ \text{in} \ Ω, \mbox{ and } \ u = g \ \text{on} \ \partial Ω, \end{align*} that related to Riesz potentials. Here, in our setting, $Ω\subset \mathbb{R}^n$, $n \ge 2$ is a bounded Reifenberg flat domain (that its boundary is sufficiently flat in sense of Reifenberg) and the small-BMO condition (small bounded me…
An eigenvalue problem for a nonlocal quasilinear anisotropic equation in fractional Orlicz Sobolev spaces without the $Δ_2$--condition
Departamento de Matemática, FCEN, Universidad de Buenos Aires（布宜诺斯艾利斯大学理学院数学系）；Instituto de Cálculo, CONICET（计算研究所，阿根廷国家科学研究委员会）；Departamento de Matemática, FCFMyN, Universidad Nacional de San Luis（圣路易斯国立大学理农医神经学院数学系）；Instituto de Matemática Aplicada San Luis, IMASL, CONICET（圣路易斯应用数学研究所，IMASL，阿根廷国家科学研究委员会）
The aim goal of this paper is to treat the following problem \begin{equation*} \left\{ \begin{array}{rcll} u_t+(-\D^s_{p}) u &=&\dyle ł\dfrac{u^{p-1}}{|x|^{ps}} & \text{ in } Ø_{T}=Ω\times (0,T), \\ u&\ge & 0 & \text{ in }\ren \times (0,T), \\ u &=& 0 & \text{ in }(\ren\setminusØ) \times (0,T), \\ u(x,0)&=& u_0(x)& \mbox{ in }Ø, \end{array}% \right. \end{equation*} where $Ω$ is a bounded domain containing the origin, $$ (-\D^s_{p})\, u(x,t):=P.V\int_{\ren} \,\dfrac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}} \,dy$$ with $1<p<N, s\in (0,1)$ and $f, u_0$ are non negative functions. The m…
We construct an efficient approach to deal with the global regularity estimates for a class of elliptic double-obstacle problems in Lorentz and Orlicz spaces. The motivation of this paper comes from the study on an abstract result in the viewpoint of the fractional maximal distributions and this work also extends some regularity results proved in \cite{PN_dist} by using the weighted fractional maximal distributions (WFMDs). We further investigate a pointwise estimates of the gradient of weak solutions via fractional maximal operators and Riesz potential of data. Moreover, in the setting of the…
In this note we complete a previous study, where we got existence results for the quasilinear elliptic equation \begin{equation*} -Δw+ V\left( \left| x\right| \right) w - w \left( Δw^2 \right)= K(|x|) g(w) \quad \text{in }\mathbb{R}^{N}, \end{equation*} with singular or vanishing continuous radial potentials $V(r)$, $K(r)$. In our previuos study we assumed, for technical reasons, that $K(r)$ was vanishing as $r \rightarrow 0$, while in the present paper we remove this obstruction. To face the problem we apply a suitable change of variables $w=f(u)$ and we find existence of non negative solutio…
We present in this note a local in time well-posedness result for the singular $2$-dimensional quasilinear generalized parabolic Anderson model equation $$ \partial_t u - a(u)Δu = g(u)ξ$$ The key idea of our approach is a simple transformation of the equation which allows to treat the problem as a semilinear problem. The analysis is done within the elementary setting of paracontrolled calculus.
We study the possibility of prescribing infinite initial values for solutions of the Evolutionary $p$-Laplace Equation in the fast diffusion case $p>2$. This expository note has been extracted from our previous work. When infinite values are prescribed on the whole initial surface, such solutions can exist only if the domain is a space-time cylinder.
This note is devoted to a small, but essential, extension of Theorem 2.1 of our recent paper [6]. The improvement is explained in Section 1 and proved in Section 2. The importance of the extension is demonstrated in Section 3 with an application to the Navier-Stokes system in critical $L_q$-spaces.
In this paper, we prove a sharp local well-posedness result for spherically symmetric solutions to quasilinear wave equations with rough initial data, when the spatial dimension is three or higher. Our approach is based on Morawetz type local energy estimates with fractional regularity for linear wave equations with variable $C^1$ coefficients, which rely on multiplier method, weighted Littlewood-Paley theory, duality and interpolation. Together with weighted linear and nonlinear estimates (including weighted trace estimates, Hardy's inequality, fractional chain rule and fractional Leibniz rul…
This note is a continuation of the work \cite{CaoXiangYan2014}. We study the following quasilinear elliptic equations \[ -Δ_{p}u-\fracμ{|x|^{p}}|u|^{p-2}u=Q(x)|u|^{\frac{Np}{N-p}-2}u,\quad\, x\in\mathbb{R}^{N}, \] where $1<p<N,0\leqμ<\left((N-p)/p\right)^{p}$ and $Q\in L^{\infty}(\R^{N})$. Optimal asymptotic estimates on the gradient of solutions are obtained both at the origin and at the infinity.
This paper studies a new gradient regularity in Lorentz spaces for solutions to a class of quasilinear divergence form elliptic equations with nonhomogeneous Dirichlet boundary conditions: \begin{align*} \begin{cases} div(A(x,\nabla u)) &= \ div(|F|^{p-2}F) \quad \text{in} \ \ Ω, \\ \hspace{1.2cm} u &=\ σ\qquad \qquad \qquad \text{on} \ \ \partial Ω. \end{cases} \end{align*} where $Ω\subset \mathbb{R}^n$ ($n \ge 2$), the nonlinearity $A$ is a monotone Carathéodory vector valued function defined on $W^{1,p}_0(Ω)$ for $p>1$ and the $p$-capacity uniform thickness condition is imposed on the compl…
We give an alternative proof for Hölder regularity for weak solutions of nonlocal elliptic quasilinear equations modelled on the fractional p-Laplacian where we replace the discrete De Giorgi iteration on a sequence of concentric balls by a continuous iteration. This work can be viewed as the nonlocal counterpart to the ideas developed by Tiziano Granucci.
In this paper, we consider the Cauchy problem for the fractional Camassa-Holm equation which models the propagation of small-but-finite amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. Using Kato's semigroup approach for quasilinear evolution equations, we prove that the Cauchy problem is locally well-posed for data in $H^{s}({\Bbb R})$, $s>{\frac{5}{2}}$.
We discuss recent advances in the theory of quasilinear equations of the type $ -Δ_{p} u = σu^{q} \; \; \text{in} \;\; \mathbb{R}^n, $ in the case $0<q< p-1$, where $σ$ is a nonnegative measurable function, or measure, for the $p$-Laplacian $Δ_{p}u= \text{div}(|\nabla u|^{p-2}\nabla u)$, as well as more general quasilinear, fractional Laplacian, and Hessian operators. Within this context, we obtain some new results, in particular, necessary and sufficient conditions for the existence of solutions $u \in \text{BMO}(\mathbb{R}^n)$, $u \in L^r_{\rm loc}(\mathbb{R}^n)$, etc., and prove an enhanced…
We study the problem of existence and uniqueness of strong solutions to a degenerate quasilinear parabolic non-Newtonian thin-film equation. Originating from a non-Newtonian Navier--Stokes system the equation is derived by lubrication theory and under the assumption that capillarity is the only driving force. The fluid's shear-thinning rheology is described by the so-called Ellis constitutive law. For flow behaviour exponents $α\geq 2$ the corresponding initial boundary value problem fits into the abstract setting of [4,Thm. 12.1]. Due to a lack of regularity this is not true for flow behaviou…
Quasilinear and Hessian type equations with exponential reaction and measure data
本文证明了在有界区域$Ω$或整个$\mathbb{R}^N$中$-Δ_pu=P(u)+μ$型方程对于$p>1$和$F_k[-u]=P(u)+μ$ with $1\leq k<\frac{N}{2}$的存在性结果，其中$μ$是正Radon测度，而$P(u)\sim e^{au^β}$ with $a>0$和$β\geq 1$ .存在的充分条件表示在$μ$的分数极大潜力。利用$μ$的一些精确Wolff势，得到了解的双边估计.根据Orlicz容量得到了必要条件。我们还在$\mathbb{R}^N$中建立了形式为$u={\bf W}_{α,p}^R[P(u)]+f$的一般Wolff势方程的存在性结果，其中0
In this work we analyze the existence of solution to the fractional quasilinear problem, \begin{equation*} \left\{ \begin{array}{rcll} (-Δ)^s u &= & |\nabla u|^{p}+ łf & \text{ in }Ω, u &=& 0 &\hbox{ in } \mathbb{R}^N\setminusΩ, u&>&0 &\hbox{ in }Ω, \end{array}% \right. \end{equation*}% where $Ω\subset \ren$ is a bounded regular domain ($\mathcal{C}^2$ is sufficient), $s\in (\frac 12, 1)$, $1<p$ and $f$ is a measurable nonnegative function with suitable hypotheses. The analysis is done separately in three cases, subcritical, $1<p<2s$, critical, $p=2s$, and supercritical, $p>2s$.
In this note we consider the action functional \[ \int_{\mathbb{R} \times ω} \left( 1 - \sqrt{ 1 - |\nabla u|^2 } + W(u) \right) \, \mathrm{d}t, \] where $W$ is a double well potential and $ω$ is a bounded domain of $\mathbb{R}^{N-1}$. We prove existence, one-dimensionality and uniqueness (up to translation) of a smooth minimizing phase transition between the two stable states $u=1$ and $u=-1$. The question of existence of at least one minimal heteroclinic connection for the non autonomous model \[ \int_{\mathbb{R}} \left( 1 - \sqrt{1-|u'|^2} + a(t) W(u) \right) \, \mathrm{d}t \] is also addre…
We extend classical results on variational inequalities with convex sets with gradient constraint to a new class of fractional partial differential equations in a bounded domain with constraint on the distributional Riesz fractional gradient, the $σ$-gradient ($0<σ<1$). We establish continuous dependence results with respect to the data, including the threshold of the fractional $σ$-gradient. Using these properties we give new results on the existence to a class of quasi-variational variational inequalities with fractional gradient constraint via compactness and via contraction arguments. Usin…
We give a very short proof of a result proved by Cappiello-Rodino-Toft on the Weyl symbol of the inverse of the Harmonic oscillator. We also extend their results to fractional powers.
James Serrin's fundamental contributions to the theory of quasilinear elliptic equations are well-known and widely appreciated. He also made less well-known contributions to the theory of quasilinear parabolic equations which we dicuss in this note. Jurgen Moser gave greatly simplified proofs of the De Giorgi-Nash regularity results for linear divergence structure elliptic and parabolic differential equations using an original iterative technique. Serrin extended Moser's techniques and applied them to the study of divergence structure quasilinear elliptic equations, and in collaboration with A…
On singular problems in nonreflexive fractional Orlicz-Sobolev spaces
Instituto de Matemática e Estatística, Universidade Federal de Goiás（戈亚斯联邦大学数学与统计研究所）；Departamento de Matemática, Universidade Federal de Roraima（罗赖马联邦大学数学系）；Departamento de Matemática, Universidade de Brasília（巴西利亚大学数学系）
In this short note, we present a Lyapunov-type inequality that corrects the recently obtained result in [M. Kirane, B. T. Torebek: A Lyapunov-type inequality for a fractional boundary value problem with Caputo-Fabrizio derivative, J. Math. Inequal. 12, 4(2018), 1005-1012].
In this paper, we study the boundedness of weak solutions to quasilinear parabolic equations of the form \[u_t - \text{div} \mathcal{A}(x,t,\nabla u) = 0, \] where the nonlinearity $\mathcal{A}(x,t,\nabla u)$ is modelled after the well studied $p$-Laplace operator. The question of boundedness has received lot of attention over the past several decades with the existing literature showing that weak solutions in either $\frac{2N}{N+2}<p<2$, $p=2$ or $2<p$ are bounded. The proof is essentially split into three cases mainly because the estimates that have been obtained in the past always included …
In this note we prove the Payne-type conjecture about the behaviour of the nodal set of least energy sign-changing solutions for the equation $-Δ_p u = f(u)$ in bounded Steiner symmetric domains $Ω\subset \mathbb{R}^N$ under the zero Dirichlet boundary conditions. The nonlinearity $f$ is assumed to be either superlinear or resonant. In the latter case, least energy sign-changing solutions are second eigenfunctions of the zero Dirichlet $p$-Laplacian in $Ω$. We show that the nodal set of any least energy sign-changing solution intersects the boundary of $Ω$. The proof is based on a moving polar…
We study a class of fractional $p$-Laplacian problems with weights which are possibly singular on the boundary of the domain. We provide existence and multiplicity results as well as characterizations of critical groups and related applications.
This paper continues the development of regularity results for quasilinear measure data problems \begin{align*} \begin{cases} -\mathrm{div}(A(x,\nabla u)) &= μ\quad \text{in} \ \ Ω, \\ \quad \quad \qquad u &=0 \quad \text{on} \ \ \partial Ω, \end{cases} \end{align*} in Lorentz and Lorentz-Morrey spaces, where $Ω\subset \mathbb{R}^n$ ($n \ge 2$), $μ$ is a finite Radon measure on $Ω$, and $A$ is a monotone Carathéodory vector valued operator acting between $W^{1,p}_0(Ω)$ and its dual $W^{-1,p'}(Ω)$. It emphasizes that this paper studies the `very singular' case $1<p \le \frac{3n-2}{2n-1}$ and th…
In this manuscript, we provide a point-wise estimate for the $3$-commutators involving fractional powers of the sub-Laplacian on Carnot groups of homogeneous dimension $Q$. This can be seen as a fractional Leibniz rule in the sub-elliptic setting. As a corollary of the point-wise estimate, we provide an $(L^{p},L^{q})\to L^{r}$ estimate for the commutator, provided that $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}-\fracα{Q}$.
We prove existence of a bounded weak solution to a degenerate quasilinear subdiffusion problem with bounded measurable coefficients that may explicitly depend on time. The kernel in the involved integro-differential operator w.r.t. time belongs to the large class of ${\cal PC}$ kernels. In particular, the case of a fractional time derivative of order less than 1 is included. A key ingredient in the proof is a new compactness criterion of Aubin-Lions type which involves function spaces defined in terms of the integro-differential operator in time. Boundedness of the solution is obtained by the …
We consider a boundary value problem driven by the fractional p-Laplacian operator with a bounded reaction term. By means of barrier arguments, we prove Hölder regularity up to the boundary for the weak solutions, both in the singular (1<p<2) and the degenerate (p>2) case.
