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Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.

11 votes
2 answers
811 views

Is it possible to obtain the inequality $\|\nabla f\|_{L^{2p}} \leq C (\|f\|_{L^\infty} \|f\...

Nirenberg's paper On elliptic PDEs claims that if a function $f$ on $\mathbb{R}^n$ tends to zero at infinity or is in $L^q$ for any $q < \infty$ then the "interpolation" inequality $$ \lVert∇ f \rVert …
Carlos Esparza's user avatar
1 vote
0 answers
157 views

Sobolev interpolation inequality for relatively compact subdomains

I was looking at Nicolaescu's Lectures on the Geometry of Manifolds (3rd edition). In Theorem 10.2.29 he presents (without proof) the following inequality: For $m \geq 1, p \geq 1, 0 < r \leq R$ there …
Carlos Esparza's user avatar
1 vote
0 answers
41 views

Deriving the general interior elliptic estimate from the compactly supported case

This is an exercise (10.3.4 in the third edition) from Nicolaescu's Lectures on the Geometry of Manifolds. Let $L$ be an elliptic differential operator of order $k$ and $1 < p < \infty$. The book prov …
Carlos Esparza's user avatar