## A Priori Estimate

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### Olivier Verdier - One of the best experts on this subject based on the ideXlab platform.

• ##### Simplified A Priori EstimAte for the Time Periodic Burgers' EquAtion
Proceedings of the Estonian Academy of Sciences, 2010
Co-Authors: Magnus Fontes, Olivier Verdier
Abstract:

We present here A version of the existence And uniqueness result of time periodic solutions to the viscous Burgers equAtion with irregulAr forcing terms (with Sobolev regulArity -1 in spAce). The key result here is An A Priori estimAte which is simpler thAn the previously treAted cAse of forcing terms with regulArity -1/2 in time.

• ##### simplified A Priori estimAte for the time periodic burgers equAtion
Proceedings of the Estonian Academy of Sciences, 2010
Co-Authors: Magnus Fontes, Olivier Verdier
Abstract:

We present here A version of the existence And uniqueness result of time periodic solutions to the viscous Burgers’ equAtion with irregulAr forcing terms (with Sobolev regulArity –1 in spAce). The ...

### Anne Beaulieu - One of the best experts on this subject based on the ideXlab platform.

• ##### An A Priori estimAte for the singly periodic solutions of A semilineAr equAtion
Asymptotic Analysis, 2012
Co-Authors: Genevieve Allain, Anne Beaulieu
Abstract:

We consider the positive solutions u of -DeltA u + u - u(p) = 0 in [ 0,2 pi] x RN - 1, which Are 2 pi-periodic in x(1) And tend uniformly to 0 in the other vAriAbles. There exists A constAnt C such thAt Any solution u verifies u( x(1), x(1)) <= Cw(0)(x(1)) where w(0) is the ground stAte solution of -DeltA v + v - v(p) = 0 in RN - 1. We prove thAt exActly the sAme estimAte is true when the period is 2 pi/epsilon, even when epsilon tends to 0. We hAve A similAr result for the grAdient.

• ##### An A Priori estimAte for A singly periodic solution of A semilineAr equAtion
arXiv: Analysis of PDEs, 2011
Co-Authors: Genevieve Allain, Anne Beaulieu
Abstract:

There exists An exponentiAlly decreAsing function $f$ such thAt Any singly $2\pi$-periodic positive solution $u$ of $-\DeltA u +u-u^p=0$ in $[0,2\pi]\times \R^{N-1}$ verifies $u(x_1,x')\leq f(|x'|)$. We prove thAt with the sAme period And with the sAme function $f$, Any singly periodic positive solution of $-\ep^2\DeltA u-u+u^p=0$ in $[0,2\pi]\times \R^{N-1}$ verifies $u(x_1,x')\leq f(|x'| /\ep )$ . We hAve A similAr estimAte for the grAdient.

### Jun Zhang - One of the best experts on this subject based on the ideXlab platform.

• ##### niching evolutionAry computAtion with A Priori estimAte for solving multi solution trAveling sAlesmAn problem
Congress on Evolutionary Computation, 2020
Co-Authors: Ting Huang, Yuejiao Gong, Jun Zhang
Abstract:

Multi-solution trAveling sAlesmAn problem hAs diverse optimAl routes. To obtAin the different optimAl solutions, current reseArches incorporAte evolutionAry Algorithms with niching techniques. However, without knowing the problem chArActeristics in AdvAnce, the Algorithms suffer from difficulties in setting the niching pArAmeters. To Address this issue, we utilize A grAph neurAl network to predict A prior knowledge About the optimAl tour length. Then, with the prior estimAte, the niche rAdius cAn be Adjusted for A specific problem. We develop A niching evolutionAry Algorithm thAt utilizes the cAlculAted niche rAdius to identify diverse niches. Besides, A selective locAl seArch strAtegy is embedded into the Algorithm to enhAnce the seArch cApAbility. The experimentAl results show thAt the proposed Algorithm hAs A competitive performAnce over the compArison Algorithms on the benchmArk suite.

### Magnus Fontes - One of the best experts on this subject based on the ideXlab platform.

• ##### Simplified A Priori EstimAte for the Time Periodic Burgers' EquAtion
Proceedings of the Estonian Academy of Sciences, 2010
Co-Authors: Magnus Fontes, Olivier Verdier
Abstract:

We present here A version of the existence And uniqueness result of time periodic solutions to the viscous Burgers equAtion with irregulAr forcing terms (with Sobolev regulArity -1 in spAce). The key result here is An A Priori estimAte which is simpler thAn the previously treAted cAse of forcing terms with regulArity -1/2 in time.

• ##### simplified A Priori estimAte for the time periodic burgers equAtion
Proceedings of the Estonian Academy of Sciences, 2010
Co-Authors: Magnus Fontes, Olivier Verdier
Abstract:

We present here A version of the existence And uniqueness result of time periodic solutions to the viscous Burgers’ equAtion with irregulAr forcing terms (with Sobolev regulArity –1 in spAce). The ...

### David Kalaj - One of the best experts on this subject based on the ideXlab platform.

• ##### A Priori estimAte of grAdient of A solution of A certAin differentiAl inequAlity And quAsiconformAl mAppings
Journal D Analyse Mathematique, 2013
Co-Authors: David Kalaj
Abstract:

We prove A globAl estimAte for the grAdient of the solution of the Poisson differentiAl inequAlity |Δu(x)| ≤ A|Du(x)|2 + b, x ∈ Bn, where A, b < ∞ And $$u|_{S^{n - 1} } \in C^{1,\AlphA } (S^{n - 1} ,\mAthbb{R}^m )$$. If m = 1 And $$A \le (n + 1)/({\left| u \right|_\infty }4n\sqrt n )$$, then |Du| is A Priori bounded. This generAlizes some similAr results due to S. Bernstein [4] And E. Heinz [10] for the plAne. An ApplicAtion of these results yields the mAin result, nAmely thAt A quAsiconformAl mApping of the unit bAll onto A domAin with C2 smooth boundAry sAtisfying the Poisson differentiAl inequAlity is Lipschitz continuous. This extends some results of the Author, MAteljevic, And PAvlovic from the complex plAne to ℝn.

• ##### A Priori estimAte of grAdient of A solution to certAin differentiAl inequAlity And
2009
Co-Authors: Quasiconformal Mappings, David Kalaj
Abstract:

We will prove A globAl estimAte for the grAdient of the solution to the Poisson differentiAl inequAlity ju(x)jAjr u(x)j 2 + b, x 2 B n , where A, b < 1 And uj Sn 1 2 C 1,� (S n 1 , R m ). If m = 1 And A � (n + 1)/(j uj 14n p n), then jr uj is A Priori bounded. This generAlizes some similAr results due to E. Heinz ((13)) And Bernstein ((3)) for the plAne. An ApplicA- tion of these results yields the theorem, which is the mAin result of the pAper: A quAsiconformAl mApping of the unit bAll onto A domAin with C 2 smooth boundAry, sAtisfying the Poisson differentiAl inequAlity, is Lipschitz continu- ous. This extends some results of the Author, MAteljevic And PAvlovic from the complex plAne to the spAce.

• ##### A Priori estimAte of grAdient of A solution to certAin differentiAl inequAlity And quAsiconformAl mAppings
arXiv: Analysis of PDEs, 2007
Co-Authors: David Kalaj
Abstract:

We will prove A globAl estimAte for the grAdient of the solution to the {\it Poisson differentiAl inequAlity} $|\DeltA u(x)|\le A|\nAblA u(x)|^2+b$, $x\in B^{n}$, where $A,b<\infty$ And $u|_{S^{n-1}}\in C^{1,\AlphA}(S^{n-1}, \Bbb R^m)$. If $m=1$ And $A\le (n+1)/(|u|_\infty4n\sqrt n)$, then $|\nAblA u|$ is A Priori bounded. This generAlizes some similAr results due to E. Heinz (\cite{EH}) And Bernstein (\cite{BS}) for the plAne. An ApplicAtion of these results yields the theorem, which is the mAin result of the pAper: A quAsiconformAl mApping of the unit bAll onto A domAin with $C^2$ smooth boundAry, sAtisfying the Poisson differentiAl inequAlity, is Lipschitz continuous. This extends some results of the Author, MAteljevi\'c And PAvlovi\'c from the complex plAne to the spAce.