In this article, we communicate with the glimpse of the proofs of global regularity results for weak solutions to a class of problems involving fractional $(p,q)$-Laplacian, denoted by $(-Δ)^{s_1}_{p}+(-Δ)^{s_2}_{q}$, for $s_2, s_1\in (0,1)$ and $1<p,q<\infty$. We also obtain the boundary Hölder continuity results for the weak solutions to the corresponding problems involving at most critical growth nonlinearities. These results are almost optimal. Moreover, we establish Hopf type maximum principle and strong comparison principle. As an application to these new results, we prove the Sobolev versus Hölder minimizer type result, which provides the multiplicity of solutions in the spirit of seminal work \cite{Brezis-Nirenberg}.
We investigate the regularity of the solutions for a class of degenerate/singular fully nonlinear nonlocal equations. In the degenerate scenario, we establish that there exists at least one viscosity solution of class $C_{loc}^{1, α}$, for some constant $α\in (0, 1)$. In addition, under suitable conditions on $σ$, we prove regularity estimates in Hölder spaces for any viscosity solution. We also examine the singular setting and prove Hölder regularity estimates for the gradient of the solutions.
The paper deals with existence and multiplicity of positive solutions to nonlocal equations with critical Hrardy-Sobolev nonlinearities and external terms. We establish the profile decomposition of the Palais-Smale sequences associated with the functional and existence of at least two positive solutions to the equation.
In this article, we deal with the global regularity of weak solutions to a class of problems involving the fractional $(p,q)$-Laplacian, denoted by $(-Δ)^{s_1}_{p}+(-Δ)^{s_2}_{q}$, for $s_2, s_1\in (0,1)$ and $1<p,q<\infty$. We establish completely new Hölder continuity results, up to the boundary, for the weak solutions to fractional $(p,q)$-problems involving singular as well as regular nonlinearities. Moreover, as applications to boundary estimates, we establish new Hopf type maximum principle and strong comparison principle in both situations.
We introduce a new class of quasi-linear parabolic equations involving nonhomogeneous degeneracy or/and singularity $$ \partial_t u=[|D u|^q+a(x,t)|D u|^s]\left(Δu+(p-2)\left\langle D^2 u\frac{D u}{|D u|},\frac{D u}{|D u|}\right\rangle\right), $$ where $1<p<\infty$, $-1<q\leq s<\infty$ and $a(x,t)\ge 0$. The motivation to investigate this model stems not only from the connections to tug-of-war like stochastic games with noise, but also from the non-standard growth problems of double phase type. According to different values of $q,s$, such equations include nonhomogeneous degeneracy or singular…
We prove existence and regularity results for free transmission problems governed by fully nonlinear elliptic equations with nonhomogeneous degeneracies.
In this paper, we study the nonhomogeneous Dirichlet problem concerning general semilinear elliptic equations in divergence form. We establish that the boundary Lipschitz regularity of solutions under some more weaker conditions on the coefficients, the boundary, the boundary function and the nonhomogeneous term. In particular, we assume that the nonhomogeneous term satisfies Dini continuity condition and Lipschitz Newtonian potential condition, which will be the optimal conditions to obtain the boundary Lipschitz regularity of solutions.
This paper is concerned with the following fractional Schrödinger equation \begin{equation*} \left\{ \begin{array}{ll} (-Δ)^{s} u+u= k(x)f(u)+h(x) \mbox{ in } \mathbb{R}^{N}\\ u\in H^{s}(\R^{N}), \, u>0 \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $s\in (0,1)$, $N> 2s$, $(-Δ)^{s}$ is the fractional Laplacian, $k$ is a bounded positive function, $h\in L^{2}(\mathbb{R}^{N})$, $h\not \equiv 0$ is nonnegative and $f$ is either asymptotically linear or superlinear at infinity.\\ By using the $s$-harmonic extension technique and suitable variational methods, we prove the exi…
In this paper, we consider the pointwise boundary Lipschitz regularity of solutions for the semilinear elliptic equations in divergence form mainly under some weaker assumptions on nonhomogeneous term and the boundary. If the domain satisfies C^{1,\text{Dini}} condition at a boundary point, and the nonhomogeneous term satisfies Dini continuous condition and Lipschitz Newtonian potential condition, then the solution is Lipschitz continuous at this point. Furthermore, we generalize this result to Reifenberg C^{1,\text{Dini}} domains.
This article deals with the study of the following singular quasilinear equation: \begin{equation*} (P) \left\{ \ -Δ_{p}u -Δ_{q}u = f(x) u^{-δ},\; u>0 \text{ in }\; \Om; \; u=0 \text{ on } \pa\Om, \right. \end{equation*} where $\Om$ is a bounded domain in $\mathbb{R}^N$ with $C^2$ boundary $\pa\Om$, $1< q< p<\infty$, $\de>0$ and $f\in L^\infty_{loc}(\Om)$ is a non-negative function which behaves like $\textnormal{dist}(x,\pa\Om)^{-\ba},$ $\ba\ge 0$ near the boundary of $\Om$. We prove the existence of a weak solution in $W^{1,p}_{loc}(\Om)$ and its behaviour near the boundary for $\ba<p$. Cons…
We prove a general perturbation theorem that can be used to obtain pairs of nontrivial solutions of a wide range of local and nonlocal nonhomogeneous elliptic problems. Applications to critical $p$-Laplacian problems, $p$-Laplacian problems with critical Hardy-Sobolev exponents, critical fractional $p$-Laplacian problems, and critical $(p,q)$-Laplacian problems are given. Our results are new even in the semilinear case $p = 2$.
In this paper, we investigate the Cauchy problem of the nonhomogeneous incompressible non-resistive MHD on $\mathbb{R}^2$ with vacuum as far field density and prove that the 2D Cauchy problem has a unique local strong solution provided that the initial density and magnetic field decay not too slow at infinity. Furthermore, if the initial data satisfy some additional regularity and compatibility conditions, the strong solution becomes a classical one.
We establish sharp global regularity results for solutions to nonhomogeneous, nonunifomrly elliptic systems with zero boundary conditions. In particular, we obtain everywhere Lipschitz continuity under borderline Lorentz assumptions on the forcing term, thus positively settling the optimality issue raised in \cite{bdms}.
The present paper deals with the nonhomogeneous incompressible asymmetric fluids equations in dimension $d= 2,3$. The aim is to prove the unique global solvability of the system with only bounded nonnegative initial density and $H^{1}$ initial velocities. We first construct the global existence of the solution with large data in 2-D. Next, we establish the existence of local in time solution for arbitrary large data and global in time for some smallness conditions in 3-D. Finally, the uniqueness of the solution is proved under quite soft assumptions about its regularity through a Lagrangian ap…
We are devoted to the study of a nonhomogeneous time-fractional Timoshenko system with frictional and viscoelastic damping terms. We are concerned with the well-posedness of the given problem. The approach relies on some functional-analysis tools, operator theory, a prori estimates, and density arguments.
In this paper we study the Dirichlet problem for the Kobayashi--Warren--Carter system. This system of parabolic PDE's models the grain boundary motion in a polycrystal with a prescribed orientation at the boundary of the domain. We obtain global existence in time of energy-dissipative solutions. The regularity of the solutions as well as the energy-dissipation property permit us to derive the steady-state problem as the asymptotic in time limit of the system. We finally study the $ω$-limit set of the solutions; we completely characterize it in the one dimensional case, showing, in particular t…
In terms of layer potential methods, this paper is devoted to study the $L^2$ boundary value problems for nonhomogeneous elliptic operators with rapidly oscillating coefficients in a periodic setting. Under a low regularity assumption on the coefficients, we establish the solvability for Dirichlet, regular and Neumann problems in a bounded Lipschitz domain, as well as, the uniform nontangential maximal function estimates and square function estimates. The main difficulty is reflected in two aspects: (i) we can not treat the lower order terms as a compact perturbation to the leading term due to…
For $2a$-order strongly elliptic operators $P$ generalizing $(-Δ)^a$, $0<a<1$, the treatment of the homogeneous Dirichlet problem on a bounded open set $Ω\subset R^n$ by pseudodifferential methods, has been extended in a recent joint work with Helmut Abels to nonsmooth settings, showing regularity theorems in $L_q$-Sobolev spaces $H_q^s$ for $1<q<\infty $, when $Ω$ is $C^{τ+1}$ with a finite $τ>2a$. Presently, we study the $L_q$-Dirichlet realizations of $P$ and $P^*$, showing invertibility or Fredholmness, finding smoothness results for the kernels and cokernels, and establishing similar resu…
In this paper, we develop some properties of the $a_{x,y}(.)$-Neumann derivative for the fractional $a_{x,y}(.)$-Laplacian operator. Therefore we prove the basic proprieties of the correspondent function spaces. In the second part of this paper, by means of Ekeland's variational principal and direct variational approach, we prove the existence of weak solutions for a nonlocal problem with nonhomogeneous Neumann and Robin boundary condition.
This paper is concerned with initial-boundary-value problems (IBVPs) for a class of nonlinear Schrödinger equations posed either on a half line $\mathbb{R}^+$ or on a bounded interval $(0, L)$ with nonhomogeneous boundary conditions. For any $s$ with $0\leq s < 5/2$ and $s \not = 3/2$, it is shown that the relevant IBVPs are locally well-posed if the initial data lie in the $L^2$--based Sobolev spaces $H^s(\mathbb{R}^+) $ in the case of the half line and in $H^s (0, L)$ on a bounded interval, provided the boundary data are selected from $H^{(2s+1)/4}_{loc} (\mathbb{R}^+)$ and $H^{(s+ 1) /2}_{l…
In this paper, we study the interplay between Orlicz-Sobolev spaces $L^{M}$ and $W^{1,M}$ and fractional Sobolev spaces $W^{s,p}$. More precisely, we give some qualitative properties of the new fractional Orlicz-Sobolev space $W^{s,M}$, where $s\in (0,1)$ and $M$ is an $N-$function. We also study a related non-local operator, which is a fractional version of the nonhomogeneous $M$-Laplace operator. As an application, we prove existence of weak solution for a non-local problem involving the new fractional $M-$Laplacian operator.
In this work we study regularity properties of solutions to fractional elliptic problems with mixed Dirichlet-Neumann boundary data when dealing with the Spectral Fractional Laplacian.
Using the Galerkin method, we obtain the unique existence of the weak solution to a time fractional wave problem, and establish some regularity estimates which reveal the singularity structure of the weak solution in time.
We study global regularity for solutions of quasilinear elliptic equations of the form $÷\A(x,u,\nabla u) = ÷\F $ in rough domains $Ω$ in $\R^n$ with nonhomogeneous Dirichlet boundary condition. The vector field $\A$ is assumed to be continuous in $u$, and its growth in $\nabla u$ is like that of the $p$-Laplace operator. We establish global gradient estimates in weighted Morrey spaces for weak solutions $u$ to the equation under the Reifenberg flat condition for $Ω$, a small BMO condition in $x$ for $\A$, and an optimal condition for the Dirichlet boundary data.
We derive some regularity estimates of the solution to a time fractional diffusion equation, that are useful for numerical analysis, and partially unravel the singularity structure of the solution with respect to the time variable.
We establish interior and up to the boundary Hölder regularity estimates for weak solutions of the Dirichlet problem for the fractional $g-$Laplacian with bounded right hand side and $g$ convex. These are the first regularity results available in the literature for integro-differential equations in the context of fractional Orlicz-Sobolev spaces.
In this note we consider regularity theory for a fractional $p$-Laplace operator which arises in the complex interpolation of the Sobolev spaces, the $H^{s,p}$-Laplacian. We obtain the natural analogue to the classical $p$-Laplacian situation, namely $C^{s+α}_{loc}$-regularity for the homogeneous equation.
We prove that for $p\ge 2$ solutions of equations modeled by the fractional $p$-Laplacian improve their regularity on the scale of fractional Sobolev spaces. Moreover, under certain precise conditions, they are in $W^{1,p}_{loc}$ and their gradients are in a fractional Sobolev space as well. The relevant estimates are stable as the fractional order of differentiation $s$ reaches $1$.
We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary bounded open set $Ω\subset\mathbb{R}^N$. Proofs combine classical abstract regularity results for parabolic equations with some new local regularity results for the associated elliptic problems.
In this article we investigate the existence and regularity of 1-d steady state fractional order diffusion equations. Two models are investigated:the Riemann-Liouville fractional diffusion equation, and the Riemann-Liouville-Caputo fractional diffusion equation. For these models we explicitly show how the regularity of the solution depends upon the right hand side (rhs) function. We also establish for which Dirichlet and Neumann boundary conditions the models are well posed.
We prove a higher regularity result for the free boundary in the obstacle problem for the fractional Laplacian via a higher order boundary Harnack inequality.
We study existence, uniqueness and regularity properties of classical solutions to viscous Hamilton-Jacobi equations with Caputo time-fractional derivative. Our study relies on a combination of a gradient bound for the time-fractional Hamilton-Jacobi equation obtained via nonlinear adjoint method and sharp estimates in Sobolev and Hölder spaces for the corresponding linear problem.
We establish new quantitative estimates for localized finite differences of solutions to the Poisson problem for the fractional Laplace operator with homogeneous Dirichlet conditions of solid type settled in bounded domains satisfying the Lipschitz cone regularity condition. We then apply these estimates to obtain (i)~regularity results for solutions of fractional Poisson problems in Besov spaces; (ii)~quantitative stability estimates for solutions of fractional Poisson problems with respect to domain perturbations; (iii)~quantitative stability estimates for eigenvalues and eigenfunctions of f…
This article addresses the regularity issue for stationary or minimizing fractional harmonic maps into spheres of order $s\in(0,1)$ in arbitrary dimensions. It is shown that such fractional harmonic maps are $C^\infty$ away from a small closed singular set. The Hausdorff dimension of the singular set is also estimated in terms of $s\in(0,1)$ and the stationarity/minimality assumption.
We prove two maximal regularity results in spaces of continuous and Hölder continuous functions, for a mixed linear Cauchy-Dirichlet problem with a fractional time derivative $\mathbb{D}_t^α$. This derivative is intended in the sense of Caputo and $α$ is taken in $(0, 2)$. In case $α= 1$, we obtain maximal regularity results for mixed parabolic problems already known in mathematica literature.
We extend the Caffarelli-Kohn-Nirenberg type partial regularity theory for the steady $5$-dimensional fractional Navier-Stokes equations with external force to the hyperdissipative setting. In our argument we use the methods of Colombo-De Lellis-Massaccesi to apply a blowup procedure adapted from work of Ladyzhenskaya-Seregin.
We are concerned with an initial boundary value problem for the nonhomogeneous heat conducting Navier-Stokes flows with non-negative density. First of all, we show that for the initial density allowing vacuum, the strong solution exists globally if the velocity satisfies the Serrin's condition. Then, under some smallness condition, we prove that there is a unique global strong solution to the 3D viscous nonhomogeneous heat conducting Navier-Stokes flows. Our method relies upon the delicate energy estimates and regularity properties of Stokes system and elliptic equation.
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.
