# Concavity of the Lagrangian phase operator and applications

Concavity of the Lagrangian phase operator and applications We study the Dirichlet problem for the Lagrangian phase operator, in both the real and complex setting. Our main result states that if $$\Omega$$ Ω is a compact domain in $${\mathbb {R}}^{n}$$ R n or $${\mathbb {C}}^n$$ C n , then there exists a solution to the Dirichlet problem with right-hand side h(x) satisfying $$|h(x)| > (n-2)\frac{\pi }{2}$$ | h ( x ) | > ( n - 2 ) π 2 and boundary data $$\varphi$$ φ if and only if there exists a subsolution. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

# Concavity of the Lagrangian phase operator and applications

, Volume 56 (4) – Jun 5, 2017
22 pages

/lp/springer_journal/concavity-of-the-lagrangian-phase-operator-and-applications-uF8bQn5AHh
Publisher
Springer Berlin Heidelberg
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
D.O.I.
10.1007/s00526-017-1191-z
Publisher site
See Article on Publisher Site

### Abstract

We study the Dirichlet problem for the Lagrangian phase operator, in both the real and complex setting. Our main result states that if $$\Omega$$ Ω is a compact domain in $${\mathbb {R}}^{n}$$ R n or $${\mathbb {C}}^n$$ C n , then there exists a solution to the Dirichlet problem with right-hand side h(x) satisfying $$|h(x)| > (n-2)\frac{\pi }{2}$$ | h ( x ) | > ( n - 2 ) π 2 and boundary data $$\varphi$$ φ if and only if there exists a subsolution.

### Journal

Calculus of Variations and Partial Differential EquationsSpringer Journals

Published: Jun 5, 2017

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