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Higher Differentiability Results for Solutions to a Class of Non-homogeneouns Elliptic Problems Under Sub-quadratic Growth Conditions by Clop, Albert; Gentile, Andrea; di Napoli, Antonia Passarelli is a scholarly article available to read on EtoBox.

What is Higher Differentiability Results for Solutions to a Class of Non-homogeneouns Elliptic Problems Under Sub-quadratic Growth Conditions about?

We prove a sharp higher differentiability result for local minimizers of functionals of the form $$\mathcal{F}\left(w,\Omega\right)=\int_{\Omega}\left[ F\left(x,Dw(x)\right)-f(x)\cdot w(x)\right]dx$$ with non-autonomous integrand $F(x,\xi)$ 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 $x\mapsto D_\xi F(x,\xi)$ 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 the assumption on the datum $f$ and on the map $x\mapsto D_\xi F(x,\xi)$, if the minimizers are assumed to be a priori bounded.

Author

Clop, Albert; Gentile, Andrea; di Napoli, Antonia Passarelli

Published

2022

Language

EN