Elliptic PDE: Variational Techniques

2012 ◽  
pp. 229-298
Author(s):  
Françoise Demengel ◽  
Gilbert Demengel
Keyword(s):  
Elliptic Pde ◽  
Variational Techniques

Periodic Problems with the Scalar p-Laplacian Resonant at any Eigenvalue via Critical Point Methods

2013 ◽  
Vol 13 (3) ◽  
Author(s):  
Sophia Th. Kyritsi ◽  
Donal O’ Regan ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Critical Point ◽  
Morse Theory ◽  
Critical Point Theory ◽  
Smooth Solution ◽  
Point Theory ◽  
Reaction Term ◽  
Variational Techniques ◽  
Periodic Problems

AbstractWe consider nonlinear periodic problems driven by the scalar p-Laplacian with a Carathéodory reaction term. Under conditions which permit resonance at infinity with respect to any eigenvalue, we show that the problem has a nontrivial smooth solution. Our approach combines variational techniques based on critical point theory with Morse theory.

Nowcasting with Data Assimilation: A Case of Global Satellite Mapping of Precipitation

2016 ◽  
Vol 31 (5) ◽  
pp. 1409-1416 ◽  
Author(s):  
Shigenori Otsuka ◽  
Shunji Kotsuki ◽  
Takemasa Miyoshi
Keyword(s):  
Data Assimilation ◽  
Space Time ◽  
Motion Vectors ◽  
The Past ◽  
Variational Techniques ◽  
Alternative Approach ◽  
Ensemble Transform Kalman Filter ◽  
Global Precipitation ◽  
Time Extrapolation ◽  
Satellite Mapping

Abstract Space–time extrapolation is a key technique in precipitation nowcasting. Motions of patterns are estimated using two or more consecutive images, and the patterns are extrapolated in space and time to obtain their future patterns. Applying space–time extrapolation to satellite-based global precipitation data will provide valuable information for regions where ground-based precipitation nowcasts are not available. However, this technique is sensitive to the accuracy of the motion vectors, and over the past few decades, previous studies have investigated methods for obtaining reliable motion vectors such as variational techniques. In this paper, an alternative approach applying data assimilation to precipitation nowcasting is proposed. A prototype extrapolation system is implemented with the local ensemble transform Kalman filter and is tested with the Japan Aerospace Exploration Agency’s Global Satellite Mapping of Precipitation (GSMaP) product. Data assimilation successfully improved the global precipitation nowcasting with the real-case GSMaP data.

scholarly journals On radial stationary solutions to a model of non-equilibrium growth

2013 ◽  
Vol 24 (3) ◽  
pp. 437-453 ◽  
Author(s):  
CARLOS ESCUDERO ◽  
ROBERT HAKL ◽  
IRENEO PERAL ◽  
PEDRO J. TORRES
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Stationary Solutions ◽  
Parabolic Partial Differential Equation ◽  
Radial Solutions ◽  
Variational Techniques ◽  
Existence And Multiplicity ◽  
Equilibrium Growth ◽  
Non Equilibrium

We present the formal geometric derivation of a non-equilibrium growth model that takes the form of a parabolic partial differential equation. Subsequently, we study its stationary radial solutions by means of variational techniques. Our results depend on the size of a parameter that plays the role of the strength of forcing. For small forcing we prove the existence and multiplicity of solutions to the elliptic problem. We discuss our results in the context of non-equilibrium statistical mechanics.

An Accelerated Method for Nonlinear Elliptic PDE

2016 ◽  
Vol 69 (2) ◽  
pp. 556-580 ◽  
Author(s):  
Hayden Schaeffer ◽  
Thomas Y. Hou
Keyword(s):  
Elliptic Pde ◽  
Nonlinear Elliptic Pde ◽  
Nonlinear Elliptic

scholarly journals Radial symmetry for an elliptic PDE with a free boundary

2021 ◽  
Vol 8 (26) ◽  
pp. 311-319
Author(s):  
Layan El Hajj ◽  
Henrik Shahgholian
Keyword(s):  
Free Boundary ◽  
Radial Symmetry ◽  
Regularity Of Solutions ◽  
Elliptic Pde ◽  
Linear Elliptic Equation ◽  
Boundary Harnack Principle ◽  
Partial Differential ◽  
Content Type ◽  
Moving Plane ◽  
Plane Technique

In this paper we prove symmetry for solutions to the semi-linear elliptic equation Δ u = f ( u )  in  B 1 , 0 ≤ u > M ,  in  B 1 , u = M ,  on  ∂ B 1 , \begin{equation*} \Delta u = f(u) \quad \text { in } B_1, \qquad 0 \leq u > M, \quad \text { in } B_1, \qquad u = M, \quad \text { on } \partial B_1, \end{equation*} where M > 0 M>0 is a constant, and B 1 B_1 is the unit ball. Under certain assumptions on the r.h.s. f ( u ) f (u) , the C 1 C^1 -regularity of the free boundary ∂ { u > 0 } \partial \{u>0\} and a second order asymptotic expansion for u u at free boundary points, we derive the spherical symmetry of solutions. A key tool, in addition to the classical moving plane technique, is a boundary Harnack principle (with r.h.s.) that replaces Serrin’s celebrated boundary point lemma, which is not available in our case due to lack of C 2 C^2 -regularity of solutions.

scholarly journals The quantum query complexity of elliptic PDE

2006 ◽  
Vol 22 (5) ◽  
pp. 691-725 ◽  
Author(s):  
Stefan Heinrich
Keyword(s):  
Query Complexity ◽  
Elliptic Pde ◽  
Quantum Query Complexity ◽  
Quantum Query

SMALL-TO-LARGE POLARON CROSSOVER IN ONE DIMENSION USING VARIATIONAL PHONON-AVERAGING TECHNIQUES

2001 ◽  
Vol 15 (13) ◽  
pp. 1923-1937 ◽  
Author(s):  
P. CHOUDHURY ◽  
A. N. DAS
Keyword(s):  
One Dimension ◽  
Density Matrix Renormalization Group ◽  
Phonon Coupling ◽  
One Dimensional ◽  
Variational Techniques ◽  
Density Matrix Renormalization ◽  
Numerical Studies ◽  
Real Picture ◽  
Analytical Approaches ◽  
Averaging Techniques

The ground-state properties of polarons in a one-dimensional chain is studied analytically within the modified Lang–Firsov (MLF) transformation using various phonon-averaging techniques. The object of the work is to examine how the analytical approaches may be improved to give rise to the real picture of polaronic properties as predicted by extensive numerical studies. The results are compared with those obtained from numerical analyses using the density matrix renormalization group (DMRG) and other variational techniques. It is observed that our results agree well with the numerical results particularly in the low and intermediate range of phonon coupling.

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