Measure data problems, lower-order terms and interpolation effects
The nonlinear PDE \[ -\mathrm{div} \;a\left(x,Du\right)+g(x,u)=f \] is studied in an open bounded subset \(\Omega\) of \(\mathbb{R}^n\), \(n\geq2\) with Dirichlet boundary condition \(u=0\) on \(\partial \Omega\). Here \(f\in L^1(\Omega)\), the vector field \(a:\Omega\times\mathbb{R}^n\rightarrow\mathbb{R}^n\) is a Carathéodory regular and statisfies the following monotonicity and \(p\)-growth conditions: \[ \nu(s^2+\left|z_1\right|^2+\left|z_2\right|^2)^{(p-2)/2}\left|z_2-z_1\right|^2\leq\left\langle a(x,z_2)-a(x,z_1),z_2-z_1\right\rangle \] and \[ \left|a(x,z)\right|\leq L\left(s^2+\left|z\right|^2\right)^{(p-1)/2} \] for every \(z_1,z_2,z\in\mathbb{R}^n\) and \(x\in\Omega\); the constants \(\nu\), \(L\) and \(s\) are s.t. \(0<\nu\leq1\leq L\) and \(s\geq0\). It is also assumed that the function \(g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}\) is a Carathéodory one such that for all \(t\) and a.e. \(x\in\Omega\) the inequality \(g(x,t)\mathrm{sgn}(t)\geq\alpha_0\left|t\right|^m\) holds for some positive constants \(\alpha_0\) and \(m\), and \(\sup_{\left|t\right|\leq\tau}g(\cdot,t)\in L^1_{\mathrm{loc}}(\Omega)\). A general approach to the regularity of solutions is suggested in the paper. It is based on estimates on the level sets of the maximal operator of the gradient of solutions in terms of the level sets of the datum \(f\). They imply local estimates in the most familiar function spaces (Lebesgue, Orlicz, Lorentz--Morrey). The results obtained are in clear accordance with the classical Gagliardo-Nirenberg interpolation theory.
Gagliardo-Nirenberg interpolation inequalities
scientific article; zbMATH DE number 6305037
scientific article