arXiv subjects
Publications and source records attributed to Filip Soudský.
The estimates of intermediate derivative play important role in the theory of partial differential equations. The modern approach to this problem is via pointwise estimates, since this allows to obtain results for various function norms. The paper aims to extend the pointwise estimate for intermediate derivative in terms of the averaging operator for higher orders derivatives.
The weak lower semicontinuity of the functional $$ F(u)=\int_Ωf(x,u,\nabla u)\, dx$$ is a classical topic that was studied thoroughly. It was shown that if the function $f$ is continuous and convex in the last variable, the functional is sequentially weakly lower semicontinuous on $W^{1,p}(Ω)$. However, the known proofs use advanced instruments of real and functional analysis. Our aim here is to present proof that can be easily understood by students familiar only with the elementary measure theory.
Given any $f$ a locally finitely piecewise affine homeomorphism of $Ω\subset \rn$ onto $Δ\subset \rn$ in $W^{1,p}$, $1\leq p < \infty$ and any $ε>0$ we construct a smooth injective map $\tilde{f}$ such that $\|f-\tilde{f}\|_{W^{1,p}(Ω,\rn)} < ε$.
We study the regularity properties of the inverse of a bilipschitz mapping $f$ belonging $W^m X_{\text{loc}}$, where $X$ is an arbitrary Banach function space. Namely, we prove that the inverse mapping $f^{-1}$ is also in $W^m X_{\text{loc}}$. Furthermore, the paper shows that the class of bilipschitz mappings in $W^m X_{\text{loc}}$ is closed with respect to composition and multiplication.
A carefully written Nirenberg's proof of the well known Gagliardo-Nirenberg interpolation inequality for intermediate derivatives in $\mathbb{R}^n$ seems, surprisingly, to be missing in literature. In our paper we shall first introduce this fundamental result and provide information about it's historical background. Afterwards we present a complete, student-friendly proof. In our proof we use the architecture of Nirenberg's proof, the proof is, however, much more detailed, containing also some differences. The reader can find a short comparison of differences and similarities in the final chapter.
Classical interpolation inequality of the type $\|u\|_{X}\leq C\|u\|_{Y}^θ\|u\|_{Z}^{1-θ}$ is well known in the case when $X$, $Y$, $Z$ are Lebesgue spaces. In this paper we show that this result may be extended by replacing norms $\|\cdot\|_{Y}$ or $\|\cdot\|_{X}$ by suitable H\" older semi-norm. We shall even prove sharper version involving weak Lorentz norm. We apply this result to prove the Gagliardo--Nirenberg inequality for a wider scale of parameters.
We generalize the classical Minkowski integral inequality to the form involving general Banach function norms.