The Experts below are selected from a list of 2019 Experts worldwide ranked by ideXlab platform
Diego Moreira - One of the best experts on this subject based on the ideXlab platform.
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Optimal Regularity for the Convex Envelope and Semiconvex Functions Related to Supersolutions of Fully Nonlinear Elliptic Equations
Communications in Mathematical Physics, 2019Co-Authors: J. Ederson M. Braga, Alessio Figalli, Diego MoreiraAbstract:In this paper we prove optimal regularity for the convex envelope of Supersolutions to general fully nonlinear elliptic equations with unbounded coefficients. More precisely, we deal with coefficients and right hand sides (RHS) in Lq with $${q \geq n}$$ . This extends the result of Caffarelli on the $${C_{loc}^{1,1}}$$ regularity of the convex envelope of Supersolutions of fully nonlinear elliptic equations with bounded RHS. Moreover, we also provide a regularity result with estimates for $${\omega}$$ -semiconvex functions that are Supersolutions to the same type of equations with unbounded RHS (i.e, RHS in $${L^{q}, q \geq n}$$ ). By a completely different method, our results here extend the recent regularity results obtained by Braga et al. (Adv Math 334:184–242, 2018) for $${q > n}$$ , as far as fully nonlinear PDEs are concerned. These results include, in particular, the apriori estimate obtained by Caffarelli et al. (Commun Pure Appl Math 38(2):209–252, 1985) on the modulus of continuity of the gradient of $${\omega}$$ -semiconvex Supersolutions (for linear equations and bounded RHS) that have a Holder modulus of semiconvexity.
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inhomogeneous hopf oleĭnik lemma and regularity of semiconvex Supersolutions via new barriers for the pucci extremal operators
Advances in Mathematics, 2018Co-Authors: Ederson J M Braga, Diego MoreiraAbstract:Abstract In this paper, we construct new barriers for the Pucci extremal operators with unbounded RHS. The geometry of these barriers is given by a Harnack inequality up to the boundary type estimate. Under the possession of these barriers, we prove a new quantitative version of the Hopf–Oleĭnik Lemma for quasilinear elliptic equations with g-Laplace type growth. Finally, we prove (sharp) regularity for ω-semiconvex Supersolutions for some nonlinear PDEs. These results are new even for second order linear elliptic equations in nondivergence form. Moreover, these estimates extend and improve a classical a priori estimate proven by L. Caffarelli, J.J. Kohn, J. Spruck and L. Nirenberg in [13] in 1985 as well as a more recent result on the C 1 , 1 regularity for convex Supersolutions obtained by C. Imbert in [33] in 2006.
Juha Kinnunen - One of the best experts on this subject based on the ideXlab platform.
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lower semicontinuity and pointwise behavior of Supersolutions for some doubly nonlinear nonlocal parabolic p laplace equations
arXiv: Analysis of PDEs, 2021Co-Authors: Agnid Banerjee, Prashanta Garain, Juha KinnunenAbstract:We discuss pointwise behavior of weak Supersolutions for a class of doubly nonlinear parabolic fractional $p$-Laplace equations which includes the fractional parabolic $p$-Laplace equation and the fractional porous medium equation. More precisely, we show that weak Supersolutions have lower semicontinuous representative. We also prove that the semicontinuous representative at an instant of time is determined by the values at previous times. This gives a pointwise interpretation for a weak supersolution at every point. The corresponding results hold true also for weak subsolutions. Our results extend some recent results in the local parabolic case and in the nonlocal elliptic case to the nonlocal parabolic case. We prove the required energy estimates and measure theoretic De Giorgi type lemmas in the fractional setting.
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some local properties of subsolutons and Supersolutions for a doubly nonlinear nonlocal parabolic p laplace equation
arXiv: Analysis of PDEs, 2020Co-Authors: Agnid Banerjee, Prashanta Garain, Juha KinnunenAbstract:We establish a local boundedness estimate for weak subsolutions to a doubly nonlinear parabolic fractional $p$-Laplace equation. Our argument relies on energy estimates and a parabolic nonlocal version of De Giorgi's method. Furthermore, by means of a new algebraic inequality, we show that positive weak Supersolutions satisfy a reverse Holder inequality. Finally, we also prove a logarithmic decay estimate for positive Supersolutions.
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Unbounded Supersolutions of some quasilinear parabolic equations: A dichotomy
Nonlinear Analysis-theory Methods & Applications, 2015Co-Authors: Juha Kinnunen, Peter LindqvistAbstract:Abstract We study unbounded “Supersolutions” of the evolutionary p -Laplace equation with slow diffusion. They are the same functions as the viscosity Supersolutions. A fascinating dichotomy prevails: either they are locally summable to the power p − 1 + n p − 0 or not summable to the power p − 2 . There is a void gap between these exponents. Those summable to the power p − 2 induce a Radon measure, while those of the other kind do not. We also sketch similar results for the Porous Medium Equation.
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Unbounded Supersolutions of Some Quasilinear Parabolic Equations: a Dichotomy
arXiv: Analysis of PDEs, 2015Co-Authors: Juha Kinnunen, Peter LindqvistAbstract:We study unbounded (viscosity) Supersolutions of the Evolutionary p-Laplace Equation in the slow diffusion case. The Supersolutions fall into two widely different classes, depending on whether they are locally summable to the power p-2 or not. Also the Porous Medium Equation is studied.
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definition and properties of Supersolutions to the porous medium equation
Crelle's Journal, 2008Co-Authors: Juha Kinnunen, Peter LindqvistAbstract:We study a wide class of Supersolutions of the porous medium equation. These Supersolutions are defined as lower semi- continuous functions obeying the comparison principle. We show that they have a spatial Sobolev gradient and give sharp summab- ility exponents. We also study pointwise behaviour.
Mouhamed Moustapha Fall - One of the best experts on this subject based on the ideXlab platform.
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Nonexistence of distributional Supersolutions of a semilinear elliptic equation with Hardy potential
Journal of Functional Analysis, 2013Co-Authors: Mouhamed Moustapha FallAbstract:Abstract In this paper we study nonexistence of non-negative distributional Supersolutions for a class of semilinear elliptic equations involving inverse-square potentials.
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sharp nonexistence results for a linear elliptic inequality involving hardy and leray potentials
Journal of Inequalities and Applications, 2011Co-Authors: Mouhamed Moustapha Fall, Roberta MusinaAbstract:We deal with nonnegative distributional Supersolutions for a class of linear elliptic equations involving inverse-square potentials and logarithmic weights. We prove sharp nonexistence results.
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sharp nonexistence results for a linear elliptic inequality involving hardy and leray potentials
arXiv: Analysis of PDEs, 2010Co-Authors: Mouhamed Moustapha Fall, Roberta MusinaAbstract:In this paper we deal with nonnegative distributional Supersolutions for a class of linear elliptic equations involving inverse-square potentials and logarithmic weights. We prove sharp nonexistence results.
Christoph Mainberger - One of the best experts on this subject based on the ideXlab platform.
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Minimal Supersolutions of convex BSDEs under constraints
Esaim: Probability and Statistics, 2016Co-Authors: Gregor Heyne, Michael Kupper, Christoph Mainberger, Ludovic TangpiAbstract:We study Supersolutions of a backward stochastic differential equation, the control processes of which are constrained to be continuous semimartingales of the form d Z = Δ d t + Γ d W . The generator may depend on the decomposition ( Δ,Γ ) and is assumed to be positive, jointly convex and lower semicontinuous, and to satisfy a superquadratic growth condition in Δ and Γ . We prove the existence of a supersolution that is minimal at time zero and derive stability properties of the non-linear operator that maps terminal conditions to the time zero value of this minimal supersolution such as monotone convergence, Fatou’s lemma and L 1 -lower semicontinuity. Furthermore, we provide duality results within the present framework and thereby give conditions for the existence of solutions under constraints.
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stability and markov property of forward backward minimal Supersolutions
Electronic Journal of Probability, 2016Co-Authors: Samuel Drapeau, Christoph MainbergerAbstract:We show stability and locality of the minimal supersolution of a forward backward stochastic differential equation with respect to the underlying forward process under weak assumptions on the generator. The forward process appears both in the generator and the terminal condition. Painleve-Kuratowski and Convex Epi-convergence are used to establish the stability. For Markovian forward processes the minimal supersolution is shown to have the Markov property. Furthermore, it is related to a time-shifted problem and identified as a viscosity supersolution of a corresponding PDE.
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Stability and Markov Property of Forward Backward Minimal Supersolutions
arXiv: Probability, 2015Co-Authors: Samuel Drapeau, Christoph MainbergerAbstract:We show stability and locality of the minimal supersolution of a forward backward stochastic differential equation with respect to the underlying forward process under weak assumptions on the generator. The forward process appears both in the generator and the terminal condition. Painlev\'e-Kuratowski and Convex Epi-convergence are used to establish the stability. For Markovian forward processes the minimal supersolution is shown to have the Markov property. Furthermore, it is related to a time-shifted problem and identified as the unique minimal viscosity supersolution of a corresponding PDE.
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Minimal Supersolutions of Convex BSDEs under Constraints
arXiv: Probability, 2013Co-Authors: Gregor Heyne, Michael Kupper, Christoph Mainberger, Ludovic TangpiAbstract:We study Supersolutions of a backward stochastic differential equation, the control processes of which are constrained to be continuous semimartingales of the form $dZ = {\Delta}dt + {\Gamma}dW$. The generator may depend on the decomposition $({\Delta},{\Gamma})$ and is assumed to be positive, jointly convex and lower semicontinuous, and to satisfy a superquadratic growth condition in ${\Delta}$ and ${\Gamma}$. We prove the existence of a supersolution that is minimal at time zero and derive stability properties of the non-linear operator that maps terminal conditions to the time zero value of this minimal supersolution such as monotone convergence, Fatou's lemma and $L^1$-lower semicontinuity. Furthermore, we provide duality results within the present framework and thereby give conditions for the existence of solutions under constraints.
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Minimal Supersolutions of BSDEs with Lower Semicontinuous Generators
arXiv: Probability, 2011Co-Authors: Gregor Heyne, Michael Kupper, Christoph MainbergerAbstract:We study the existence and uniqueness of minimal Supersolutions of backward stochastic differential equations with generators that are jointly lower semicontinuous, bounded below by an affine function of the control variable and satisfy a specific normalization property.
Vy Khoi Le - One of the best experts on this subject based on the ideXlab platform.
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on variational inequalities with maximal monotone operators and multivalued perturbing terms in sobolev spaces with variable exponents
Journal of Mathematical Analysis and Applications, 2012Co-Authors: Vy Khoi LeAbstract:Abstract We are concerned in this paper with variational inequalities of the form: { 〈 A ( u ) , v − u 〉 + 〈 F ( u ) , v − u 〉 ⩾ 〈 L , v − u 〉 , ∀ v ∈ K , u ∈ K , where A is a maximal monotone operator, F is an integral multivalued lower order term, and K is a closed, convex set in a Sobolev space of variable exponent. We study both coercive and noncoercive inequalities. In the noncoercive case, a sub-supersolution approach is followed to obtain the existence and some other qualitative properties of solutions between sub- and Supersolutions.
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on a sub supersolution method for variational inequalities with leray lions operators in variable exponent spaces
Nonlinear Analysis-theory Methods & Applications, 2009Co-Authors: Vy Khoi LeAbstract:Abstract In this paper, we consider a sub–supersolution method for variational inequalities with Leray–Lions operators in Sobolev spaces with variable exponents. Existence and qualitative properties of solutions between sub and Supersolutions are established.
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on a sub supersolution method for the prescribed mean curvature problem
Czechoslovak Mathematical Journal, 2008Co-Authors: Vy Khoi LeAbstract:The paper is about a sub-supersolution method for the prescribed mean curvature problem. We formulate the problem as a variational inequality and propose appropriate concepts of sub-and Supersolutions for such inequality. Existence and enclosure results for solutions and extremal solutions between sub-and Supersolutions are established.
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Some general concepts of sub- and Supersolutions for nonlinear elliptic problems
Topological Methods in Nonlinear Analysis, 2006Co-Authors: Vy Khoi Le, Klaus SchmittAbstract:We propose general and unified concepts of sub- Supersolutions for boundary value problems that encompass several types of boundary conditions for nonlinear elliptic equations and variational inequalities. Various, by now classical, sub- and supersolution existence and comparison results are covered by the general theory presented here.
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sub supersolution method for quasilinear parabolic variational inequalities
Journal of Mathematical Analysis and Applications, 2004Co-Authors: Siegfried Carl, Vy Khoi LeAbstract:This paper is about a systematic attempt to apply the sub-supersolution method to parabolic variational inequalities. We define appropriate concepts of sub-Supersolutions and derive existence, comparison, and extremity results for such inequalities.
