We study the higher fractional differentiability properties of the gradient of the solutions to variational obstacle problems of the form \begin{gather*} \min \biggl\{ \int_Ω F(x,w,Dw) d x \ : \ w \in \mathcal{K}_ψ(Ω) \biggr\}, \end{gather*} with $F$ double phase functional of the form \begin{equation*} F(x,w,z)=b(x,w)(|z|^p+a(x)|z|^q), \end{equation*} where $Ω$ is a bounded open subset of $\mathbb{R}^n$, $ψ\in W^{1,p}(Ω)$ is a fixed function called \textit{obstacle} and $\mathcal{K}_ψ(Ω)= \{ w \in W^{1,p}(Ω) : w \geq ψ\ \text{a.e. in} \ Ω\}$ is the class of admissible functions. Assuming that the gradient of the obstacle belongs to a suitable Besov space, we are able to prove that the gradient of the solution preserves some fractional differentiability property.
In this paper we study the Eikonal equation in a bounded planar domain. We prove the equivalence among optimal Besov regularity, the finiteness of every entropy production and the validity of a kinetic formulation.
We consider the inverse Stefan type free boundary problem, where information on the boundary heat flux and density of the sources are missing and must be found along with the temperature and the free boundary. We pursue optimal control framework where boundary heat flux, density of sources, and free boundary are components of the control vector. The optimality criteria consists of the minimization of the $L_2$-norm declinations of the temperature measurements at the final moment, phase transition temperature, and final position of the free boundary. We prove the Frechet differentiability in Be…
We establish some higher differentiability results of integer and fractional order for solution to non-autonomous obstacle problems of the form \begin{equation*} \min \left\{\int_Ωf(x, Dv(x))\,:\, v\in \mathcal{K}_ψ(Ω)\right\}, \end{equation*} where the function $f$ satisfies $p-$growth conditions with respect to the gradient variable, for $1<p<2$, and $\mathcal{K}_ψ(Ω)$ is the class of admissible functions $v\in u_0+W^{1, p}_0(Ω)$ such that $v\geψ$ a. e. in $Ω$, where $u_0\in W^{1,p}(Ω)$ is a fixed boundary datum. Here we show that a Sobolev or Besov-Lipschitz regularity assumption on the gra…
We here establish the higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions. We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality of the form \begin{equation*} \displaystyle\int_Ω \langle \mathcal{A}(x,Du) ,D(φ-u) \rangle dx \geq 0 \qquad \forall φ\in \mathcal{K}_ψ(Ω), \end{equation*} where $Ω$ is a bounded open subset of $\mathbb{R}^n$, $ψ\in W^{1,p}(Ω)$ is a fixed function called \textit{obstacle} and $\mathcal{K}_ψ(Ω)= \{ w \in W^{1,p}(Ω) : w \geq ψ\ \text{a.e. in} \ Ω\}$ is the…
We establish the higher fractional differentiability of bounded minimizers to a class of obstacle problems with non-standard growth conditions of the form \begin{gather*} \min \biggl\{ \displaystyle\int_Ω F(x,Dw)dx \ : \ w \in \mathcal{K}_ψ(Ω) \biggr\}, \end{gather*} where $Ω$ is a bounded open set of $\mathbb{R}^n$, $n \geq 2$, the function $ψ\in W^{1,p}(Ω)$ is a fixed function called \textit{obstacle} and $\mathcal{K}_ψ(Ω) := \{ w \in W^{1,p}(Ω) : w \geq ψ\ \text{a.e. in} \ Ω\}$ is the class of admissible functions. If the obstacle $ψ$ is locally bounded, we prove that the gradient of soluti…
In this paper we prove a higher differentiability result for the solutions to a class of obstacle problems in the form \begin{equation*} \label{obst-def0} \min\left\{\int_ΩF(x,Dw) dx : w\in \mathcal{K}_ψ(Ω)\right\} \end{equation*} where $ψ\in W^{1,p(x)}(Ω)$ is a fixed function called obstacle and $\mathcal{K}_ψ=\{w \in W^{1,p(x)}_{0}(Ω)+u_0: w \ge ψ\,\, \textnormal{a.e. in $Ω$}\}$ is the class of the admissible functions, for a suitable boundary value $ u_0 $. We deal with a convex integrand $F$ which satisfies the $p(x)$-growth conditions \begin{equation*}\label{growth}|ξ|^{p(x)}\le F(x,ξ)\le…
In this paper, we consider the well-posedness of the inhomogeneous nonlinear biharmonic Schrödinger equation with spatial inhomogeneity coefficient $K(x)$ behaves like $\left|x\right|^{-b}$ for $0<b<\min \left\{\frac{N}{2},4\right\} $. We show the local well-posedness in the whole $H^s$-subcritical case, with $0<s\le2$. The difficulties of this problem come from the singularity of $K(x)$ and the lack of differentiability of the nonlinear term. To resolve this, we derive the bilinear Strichartz's type estimates for the nonlinear biharmonic Schrödinger equations in Besov spaces.
The directional differentiability of the solution map of obstacle type quasi-variational inequalities (QVIs) with respect to perturbations on the forcing term is studied. The classical result of Mignot is then extended to the quasi-variational case under assumptions that allow multiple solutions of the QVI. The proof involves selection procedures for the solution set and represents the directional derivative as the limit of a monotonic sequence of directional derivatives associated to specific variational inequalities. Additionally, estimates on the coincidence set and several simplifications …
Motivated by applications to congested traffic problems, we establish higher integrability results for the gradient of local weak solutions to the strongly degenerate or singular elliptic PDE $-\mathrm{div}\left((\vert\nabla u\vert-1)_{+}^{q-1}\frac{\nabla u}{\vert\nabla u\vert}\right)=f$, $\mathrm{in}\,\,Ω$, where $Ω$ is a bounded domain in $\mathbb{R}^{n}$ for $n\geq2$, $1<q<\infty$ and $\left(\,\cdot\,\right)_{+}$ stands for the positive part. We assume that the datum $f$ belongs to a suitable Sobolev or Besov space. The main novelty here is that we deal with the case of subquadratic growth…
We focus on elliptic quasi-variational inequalities (QVIs) of obstacle type and prove a number of results on the existence of solutions, directional differentiability and optimal control of such QVIs. We give three existence theorems based on an order approach, an iteration scheme and a sequential regularisation through partial differential equations. We show that the solution map taking the source term into the set of solutions of the QVI is directionally differentiable for general data and locally Hadamard differentiable obstacle mappings, thereby extending in particular the results of our p…
In this article we investigate the temporal regularity of strong solutions to the stochastic $p$-\com{L}aplace system in the degenerate setting, $p \in [2,\infty)$, driven by a multiplicative nonlinear stochastic forcing. We establish $1/2$ time differentiability in an expontential Besov-Orlicz space for the solution process $u$. Furthermore, we prove $1/2$ time differentiability of the nonlinear gradient $|\nabla u|^{\frac{p-2}{2}} \nabla u$ in a Nikolskii space.
In this paper we prove higher regularity for 2m-th order parabolic equations with general boundary conditions. This is a kind of maximal L_p-L_q regularity with differentiability, i.e. the main theorem is isomorphism between the solution space and the data space using Besov and Triebel--Lizorkin spaces. The key is compatibility conditions for the initial data. We are able to get a unique smooth solution if the data satisfying compatibility conditions are smooth.
We analyze the state constrained inverse Stefan type parabolic free boundary problem as an optimal control problem in the Sobolev-Besov spaces framework. Boundary heat flux, density of heat sources, and free boundary are components of the control vector. Cost functional is the sum of the $L_2$-norm declinations of the temperature measurement at the final moment, the phase transition temperature, the final position of the free boundary, and the penalty term, taking into account the state constraint on the temperature. We prove the existence of optimal control, Frechet differentiability, and opt…
In this note we are concerned with interior regularity properties of the $p$-Poisson problem $Δ_p(u)=f$ with $p>2$. For all $0<λ\leq 1$ we constuct right-hand sides $f$ of differentiability $-1+λ$ such that the (Besov-) smoothness of corresponding solutions $u$ is essentially limited to $1+λ/ (p-1)$. The statements are of local nature and cover all integrability parameters. They particularly imply the optimality of a shift theorem due to Savaré [J. Funct. Anal. 152:176-201, 1998], as well as of some recent Besov regularity results of Dahlke et al. [Nonlinear Anal. 130:298-329, 2016]. Keywords:…
We prove Lipschitz continuity results for solutions to a class of obstacle problems under standard growth conditions of $p$-type, $p \geq 2$. The main novelty is the use of a linearization technique going back to [28] in order to interpret our constrained minimizer as a solution to a nonlinear elliptic equation, with a bounded right-hand side. This leads us to start a Moser iteration scheme which provides the $L^\infty$ bound for the gradient. The application of a recent higher differentiability result [24] allows us to simplify the procedure of the identification of the Radon measure in the l…
We establish some higher differentiability results for solution to non-autonomous obstacle problems of the form \begin{equation*} \min \left\{\int_Ωf\left(x, Dv(x)\right)dx\,:\, v\in \mathcal{K}_ψ(Ω)\right\}, \end{equation*} where the function $f$ satisfies $p-$growth conditions with respect to the gradient variable, for $1<p<2$, and $\mathcal{K}_ψ(Ω)$ is the class of admissible functions. Here we show that, if the obstacle $ψ$ is bounded, then a Sobolev regularity assumption on the gradient of the obstacle $ψ$ transfers to the gradient of the solution, provided the partial map $x\mapsto D_ξf(…
In this paper we consider a class of obstacle problems of the type %\begin{equation*} %\int_Ω\left<A(x, Du), D(φ-u)\right> \, \dx\ge0\qquad\forall %φ\in W^{1,q}(Ω) \quad {\mathrm{s.t.}} \quad φ\ge ψ%\end{equation*} \begin{equation*} \min \left\{\int_Ωf(x, Dv)\, \dx\,:\, v\in \mathcal{K}_ψ(Ω)\right\} \end{equation*} where $ψ$ is the obstacle, $\mathcal{K}_ψ(Ω)=\{v\in u_0+W^{1, p}_{0}(Ω, \R): v\geψ\text{ a.e. in }Ω\}$, with $u_0 \in W^{1,p}(Ω)$ a fixed boundary datum, the class of the admissible functions and the integrand $f(x, Dv)$ satisfies non standard $(p,q)$-growth conditions. \\ We prove …
We study the shape differentiability of a cost function for the flow of an incompressible viscous fluid of power-law type. The fluid is confined to a bounded planar domain surrounding an obstacle. For smooth perturbations of the shape of the obstacle we express the shape gradient of the cost function which can be subsequently used to improve the initial design.
We consider optimal control problems for a wide class of bilateral obstacle problems where the control appears in a possibly nonlinear source term. The non-differentiability of the solution operator poses the main challenge for the application of efficient optimization methods and the characterization of Bouligand generalized derivatives of the solution operator is essential for their theoretical foundation and numerical realization. In this paper, we derive specific elements of the Bouligand generalized differential if the control operator satisfies natural monotonicity properties. We constru…
