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On inequalities between norms of partial derivatives on convex domains

Alexander Plakhov, Vladimir Protasov

Abstract

We consider inequalities between $L_p$-norms of partial derivatives, $p\in [1,+\infty]$, for bivariate concave functions on a convex domain that vanish on the boundary. Can the ratio between those norms be arbitrarily large? If not, what is the upper bound? We show that for $p=1$, the ratio is always bounded and find sharp estimates, while for $p>1$, the answer depends on the geometry of the domain.

On inequalities between norms of partial derivatives on convex domains

Abstract

We consider inequalities between -norms of partial derivatives, , for bivariate concave functions on a convex domain that vanish on the boundary. Can the ratio between those norms be arbitrarily large? If not, what is the upper bound? We show that for , the ratio is always bounded and find sharp estimates, while for , the answer depends on the geometry of the domain.
Paper Structure (1 section, 16 theorems, 50 equations, 4 figures)

This paper contains 1 section, 16 theorems, 50 equations, 4 figures.

Table of Contents

  1. Acknowledgements

Key Result

Theorem 1

For every $u\in {\mathcal{U}}_{\, \Omega}$, we have where $w_{x}, w_y$ are the sidelengths of the rectangle circumscribed around $\Omega$ with the sides parallel to the $x$ and $y$-axes respectively. Inequality (eq.5-1) becomes an equality if and only if $\Omega$ contains a vertical segment with ends on the horizontal sides of the circumscribed recta

Figures (4)

  • Figure 1: In figure (a), both vertical lines of support to $\Omega$ are angular, while in figure (b), both lines are not angular: the left line is tangent, and the right one is half-tangent to $\Omega$.
  • Figure 2: Here $\Omega$ is the upper half-circle, and $\Omega'$ contains $\Omega$ and is bounded by the dashed broken line on the right hand side.
  • Figure 3: The tangent plane to the graph of $u$.
  • Figure 4: In figure (a), $\partial\Omega$ does not contain vertical segments, while in figure (b), $\partial\Omega$ contains the vertical segment $x=a, \ y \in [\varphi_1(a),\, \varphi_2(a)]$.

Theorems & Definitions (23)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Lemma 1
  • Proposition 1
  • Theorem 2
  • Theorem 3
  • Remark 3
  • Corollary 1
  • Corollary 2
  • ...and 13 more