Lorentz estimate with a variable power for parabolic obstacle problems with non-standard growths
In the paper under review, the authors prove a global Calderón-Zygmund type estimate for weak solutions to the \(p(x,t)\)-Laplacian type parabolic obstacle problems in Lorentz spaces. More precisely, let \(u:\Omega_T:=(a,a+T)\times \Omega\rightarrow \mathbb{R}\) be the weak solution (in the sense of variational inequality) to the parabolic obstacle problem \[ u \geq \psi\text{ in } \Omega_T, \] \[ \partial_t u - \text{div } \mathbf{a}(t,x, Du) + \text{div } ( |\mathbf{f}|^{p(t,x)-2}\mathbf{f})\geq 0 \text{ in } \Omega_T, \] \[ \partial_t u - \text{div } \mathbf{a}(t,x, Du) + \text{div } ( |\mathbf{f}|^{p(t,x)-2}\mathbf{f})= 0 \text{ in } \Omega_T, \] \[ u(0,x)=u_a(x) \text{ on } \Omega, \] \[ u(t,x)=0 \text{ on } (a,a+T)\times \partial\Omega, \] where \(\Omega\subset \mathbb{R}^n\) \((n\geq 2)\) is a bounded domain, \(\psi:\Omega_T\rightarrow \mathbb{R}\) is a given obstacle, \(a\in \mathbb{R}\) and \(\mathbf{a}\) is a Carathèodory nonlinearity which satisfies classical ellipticity conditions with \(p(t,x)\)-growth, \(\frac{2n}{n+2}<\gamma_1:=\inf_{\Omega_T}p(t,x)\leq \sup_{\Omega_T}p(t,x)<\infty\). The authors prove that if the variable exponent \(p(t,x)\) satisfies a strong type log-Hölder continuity, the associated nonlinearity \(\mathbf{a}(t,x,\xi)\) is measurable in time and has small BMO semi-norm in space, the underlying domain \(\Omega\) is quasiconvex, then the weak solution satisfies the gradient estimate in the Lorentz spaces \(L^{(\gamma,s)}(\Omega_T)\): \[ |\partial_t\psi|^{\gamma'_1 q(t,x)}, |D\psi|^{p(t,x)q(t,x)}, |\mathbf{f}|^{p(t,x)q(t,x)}\in L^{(\gamma, s)}(\Omega_T)\Longrightarrow |Du|^{p(t,x)q(t,x)}\in L^{(\gamma,s)}(\Omega_T) \] for \(\gamma\in [1,\infty)\), \(s\in (0,\infty]\), and \(1<\inf_{\Omega_T}q(t,x)\leq \sup _{\Omega_T}q(t,x)<\infty\) with a log-Hölder continuity assumption. The authors aim to find the minimal regularity assumptions on the nonlinearities and the boundary of domains such that the above gradient estimate holds true.
scientific article; zbMATH DE number 6973896
scientific article