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43
Lack of compactness in the 2d critical sobolev embedding, the general case
 COMPTES RENDUS MATHEMATIQUE
, 2012
"... Abstract. This paper is devoted to the description of the lack of compactness of H 1 rad (R2)intheOrliczspace. Ourresultisexpressedintermsoftheconcentrationtype examples derived by P.L. Lions in [30]. The approach that we adopt to establish this characterization is completely different from the met ..."
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Cited by 17 (11 self)
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Abstract. This paper is devoted to the description of the lack of compactness of H 1 rad (R2)intheOrliczspace. Ourresultisexpressedintermsoftheconcentrationtype examples derived by P.L. Lions in [30]. The approach that we adopt to establish this characterization is completely different from the methods used in the study of the lack of compactness of Sobolev embedding in Lebesgue spaces and take into account the variational aspect of Orlicz spaces. We also investigate the feature of the solutions of non linear wave equation with exponential growth, where the Orlicz norm plays a decisive role.
Double logarithmic inequality with a sharp constant
 Proc. Amer. Math. Soc
"... Abstract. We prove a Log Log inequality with a sharp constant. We also show that the constant in the Log estimate is “almost ” sharp. These estimates are applied to prove a MoserTrudinger type inequality for solutions of a 2D wave equation. 1. Introduction and statement of the results By the Sobole ..."
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Cited by 14 (10 self)
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Abstract. We prove a Log Log inequality with a sharp constant. We also show that the constant in the Log estimate is “almost ” sharp. These estimates are applied to prove a MoserTrudinger type inequality for solutions of a 2D wave equation. 1. Introduction and statement of the results By the Sobolev embedding theorem, it is well known that the Sobolev space H 1 (R 2) is embedded in all Lebesgue spaces L p (R 2) for 2 ≤ p < + ∞ but not in L ∞ (R 2). Moreover, H 1 functions are in a socalled Orlicz space i.e their exponential powers are integrable functions. Precisely, we have the following MoserTrudinger inequality (see [1], [10], [11]).
An optimal transportation problem with a cost given by the euclidean distance plus import/export taxes on the boundary
"... Abstract. In this paper we analyze a mass transportation problem in a bounded domain in which there is the possibility of import/export mass across the boundary paying a tax fee in addition to the transport cost that is assumed to be given by the Euclidean distance. We show a general duality argumen ..."
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Cited by 10 (10 self)
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Abstract. In this paper we analyze a mass transportation problem in a bounded domain in which there is the possibility of import/export mass across the boundary paying a tax fee in addition to the transport cost that is assumed to be given by the Euclidean distance. We show a general duality argument and for the dual problem we find a Kantorovich potential as the limit as p → ∞ of solutions to p−Laplacian type problems with non linear boundary conditions. In addition, we show that this limit encodes all the relevant information for our problem, it provides the masses that are exported and imported from the boundary and also allows us the construction of an optimal transport plan. Finally we show that the arguments can be addapted to deal with the case in which the amount of mass that can be exported/imported is bounded by prescribed functions. 1. Introduction. Mass transport problems have been widely considered in the literature recently. This is due not only to its relevance for applications but also for the novelty of the methods needed for its solution. The origin of such problems dates back to a work from 1781 by
Structural properties of Bilateral Grand Lebesque Spaces
 TURK J MATH
, 2010
"... In this paper we study the multiplicative, tensor, Sobolev and convolution inequalities in certain Banach spaces, the socalled Bilateral Grand Lebesque Spaces. We also give examples to show the sharpness of these inequalities when possible. ..."
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Cited by 9 (6 self)
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In this paper we study the multiplicative, tensor, Sobolev and convolution inequalities in certain Banach spaces, the socalled Bilateral Grand Lebesque Spaces. We also give examples to show the sharpness of these inequalities when possible.
SYMMETRIZATION AND SHARP SOBOLEV INEQUALITIES IN METRIC SPACES
, 2008
"... We derive sharp Sobolev inequalities for Sobolev spaces on metric spaces. In particular, we obtain new sharp Sobolev embeddings and FaberKrahn estimates for Hörmander vector fields. ..."
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Cited by 7 (5 self)
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We derive sharp Sobolev inequalities for Sobolev spaces on metric spaces. In particular, we obtain new sharp Sobolev embeddings and FaberKrahn estimates for Hörmander vector fields.
MULTIPLE WEIGHT RIESZ AND FOURIER TRANSFORMS IN BILATERAL ANISOTROPIC GRAND LEBESGUE SPACES
, 2012
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Symmetrization and mass comparison for degenerate nonlinear parabolic and related elliptic equations
 ADV. NONLINEAR STUD
, 2005
"... We consider the solutions to various nonlinear parabolic equations and their elliptic counterparts and prove comparison results based on two main tools, symmetrization and mass concentration comparison. The work focuses on equations like the porous medium equation, the filtration equation and the p ..."
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Cited by 6 (3 self)
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We consider the solutions to various nonlinear parabolic equations and their elliptic counterparts and prove comparison results based on two main tools, symmetrization and mass concentration comparison. The work focuses on equations like the porous medium equation, the filtration equation and the pLaplacian equation. The results will be used in a companion work in combination with a detailed knowledge of special solutions to obtain sharp a priori bounds and decay estimates for wide classes of solutions of those equations.
AN ASYMMETRIC AFFINE PÓLYA–SZEGÖ PRINCIPLE
, 2009
"... An affine rearrangement inequality is established which strengthens and implies the recently obtained affine Pólya–Szegö symmetrization principle for functions on R n. Several applications of this new inequality are derived. In particular, a sharp affine logarithmic Sobolev inequality is establishe ..."
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Cited by 6 (1 self)
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An affine rearrangement inequality is established which strengthens and implies the recently obtained affine Pólya–Szegö symmetrization principle for functions on R n. Several applications of this new inequality are derived. In particular, a sharp affine logarithmic Sobolev inequality is established which is stronger than its classical Euclidean counterpart.
MASS TRANSPORT PROBLEMS FOR THE EUCLIDEAN DISTANCE OBTAINED AS LIMITS OF p−LAPLACIAN TYPE PROBLEMS WITH OBSTACLES
"... Abstract. In this paper we analyze a mass transportation problem that consists in moving optimally (paying a transport cost given by the Euclidean distance) an amount of a commodity larger or equal than a fixed one to fulfil a demand also larger or equal than a fixed one, with the obligation of payi ..."
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Cited by 4 (4 self)
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Abstract. In this paper we analyze a mass transportation problem that consists in moving optimally (paying a transport cost given by the Euclidean distance) an amount of a commodity larger or equal than a fixed one to fulfil a demand also larger or equal than a fixed one, with the obligation of paying an extra cost of −g1(x) for extra production of one unit at location x and an extra cost of g2(y) for creating one unit of demand at y. The extra amounts of mass (commodity/demand) are unknowns of the problem. Our approach to this problem is by taking the limit as p → ∞ to a double obstacle problem (with obstacles g1, g2) for the p−Laplacian. In fact, under a certain natural constraint on the extra costs (that is equivalent to impose that the total optimal cost is bounded) we prove that this limit gives the extra material and extra demand needed for optimality and a Kantorovich potential for the mass transport problem involved. We also show that this problem can be interpreted as an optimal mass transport problem in which one can make the transport directly (paying a cost given by the Euclidean distance) or may hire a courier that cost g2(y) − g1(x) to pick up a unit of mass at y and deliver it to x. For this different interpretation we provide