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 gradient of the obstacle $ψ$ transfers to the gradient of the solution, provided the partial map $x\mapsto D_ξf(x,ξ)$ belongs to a suitable Sobolev or Besov space. The novelty here is that we deal with subquadratic growth conditions with respect to the gradient variable, i. e. $f(x, ξ)\approx a(x)|ξ|^p$ with $1<p<2,$ and where the map $a$ belongs to a Sobolev or Besov-Lipschitz space.
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 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…
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…
We investigate under which assumptions the flow associated to autonomous planar vector fields inherits the Sobolev or BV regularity of the vector field. We consider nearly incompressible and divergence-free vector fields, taking advantage in both cases of the underlying Hamiltonian structure. Finally we provide an example of an autonomous planar Sobolev divergence-free vector field, such that the corresponding regular Lagrangian flow has no bounded variation.
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 …
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…
We study some regularity issues for solutions of non-autonomous obstacle problems with $(p,q)$-growth. Under suitable assumptions, our analysis covers the main models available in the literature.
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(…
Consider the Navier-Stokes flow past a rotating obstacle with a general time-dependent angular velocity and a time-dependent outflow condition at infinity -- sometimes called an Oseen condition. By a suitable change of coordinates the problem is transformed to an non-autonomous problem with unbounded drift terms on a fixed exterior domain $Ω\subset \R^d$. It is shown that the solution to the linearized problem is governed by a strongly continuous evolution system $\{T_Ω(t,s)\}_{t\geq s\geq0}$ on $L^p_σ(Ω)$ for $1<p<\infty$. Moreover, $L^p$-$L^q$ smoothing properties and gradient estimates of $…
We prove a sharp higher differentiability result for local minimizers of functionals of the form $$\mathcal{F}\left(w,Ω\right)=\int_Ω\left[ F\left(x,Dw(x)\right)-f(x)\cdot w(x)\right]dx$$ with non-autonomous integrand $F(x,ξ)$ which is convex with respect to the gradient variable, under $p$-growth conditions, with $1<p<2$. The main novelty here is that the results are obtained assuming that the partial map $x\mapsto D_ξF(x,ξ)$ has weak derivatives in some Lebesgue space $L^q$ and the datum $f$ is assumed to belong to a suitable Lebesgue space $L^r$. We also prove that it is possible to weaken …
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…
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 consider a class of integral functionals with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x variable belongs to a suitable Sobolev space W^{1;q}. We prove a result of higer differentiability for the minimizers. We also infer a result of Lipschitz regularity of minimizers if q > n, and a result of higher integrability for the gradient if q = n. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.
We establish local higher integrability and differentiability results for minimizers of variational integrals $$ \mathfrak{F}(v,Ω) = \int_Ω /! F(Dv(x)) \, dx $$ over $W^{1,p}$--Sobolev mappings $u \colon Ω\subset {\mathbb R}^n \to {\mathbb R}^N$ satisfying a Dirichlet boundary condition. The integrands $F$ are assumed to be autonomous, convex and of $(p,q)$ growth, but are otherwise not subjected to any further structure conditions, and we consider exponents in the range $1<p \leq q < p^{\ast}$, where $p^{\ast}$ denotes the Sobolev conjugate exponent of $p$.
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…
