By Stefano Bianchini, Eric A. Carlen, Alexander Mielke, Cédric Villani

This quantity collects the notes of the CIME direction "Nonlinear PDE’s and functions" held in Cetraro (Italy) on June 23–28, 2008. It comprises 4 sequence of lectures, introduced via Stefano Bianchini (SISSA, Trieste), Eric A. Carlen (Rutgers University), Alexander Mielke (WIAS, Berlin), and Cédric Villani (Ecole Normale Superieure de Lyon). They provided a large evaluation of far-reaching findings and fascinating new advancements bearing on, particularly, optimum delivery idea, nonlinear evolution equations, sensible inequalities, and differential geometry. A sampling of the most themes thought of the following comprises optimum shipping, Hamilton-Jacobi equations, Riemannian geometry, and their hyperlinks with sharp geometric/functional inequalities, variational tools for learning nonlinear evolution equations and their scaling homes, and the metric/energetic idea of gradient flows and of rate-independent evolution difficulties. The ebook explores the elemental connections among all of those subject matters and issues to new examine instructions in contributions via prime specialists in those fields.

**Read Online or Download Nonlinear PDE's and Applications: C.I.M.E. Summer School, Cetraro, Italy 2008, Editors: Luigi Ambrosio, Giuseppe Savaré PDF**

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**Additional info for Nonlinear PDE's and Applications: C.I.M.E. Summer School, Cetraro, Italy 2008, Editors: Luigi Ambrosio, Giuseppe Savaré**

**Example text**

This is the potential of g. Let PM be the class of functions deﬁned by PM = { G ∗ g : g ∈ EM }. For any function ϕ ∈ E ∗ , we deﬁne DM (ϕ) = inf u∈PM 1 2 R2 |∇ϕ − ∇u|2 dx (53) provided the distribution gradient of ϕ is locally square integrable, and we deﬁne DM (ϕ) to be inﬁnite otherwise. The Legendre transform of JM will consist of two parts: One is DM , and the other is the functional EM deﬁned as follows: For all ϕ ∈ E ∗ , EM (ϕ) = 1 2 R2 ∇ ϕ− M ln hM 8π 2 dx + R2 ϕhM dx + M2 , 8π (54) provided the distribution gradient of ϕ is locally square integrable, and we deﬁne EM (ϕ) to be inﬁnite otherwise.

A. Carlen translates and scalings. After all, we know from Lieb’s Theorem that the HLS is saturated by, and only by, multiples, translates and scalings of 1 1 + |x|2 d/p(λ) . Going through the limiting argument, one arrives at the conclusion, checkable by direct calculation, that hM does saturate the log HLS inequality. However, whenever one proves an inequality by such a limiting process, it is not a priori clear that additional cases of inequality do not “sneak in” through the limit. Thus, a separate proof is required for the statement that there are not other non-trivial cases of equality, apart from translates and scaling of hM , where by a non-trivial case of equality, we mean one in which both sides are ﬁnite.

To eﬃciently state the main result, we ﬁrst make some deﬁnitions. First, let us use the convenient notation − 1 2π R2 ln |x − y|g(y)dy = G ∗ g, (52) for non-negative g ∈ E, or more generally, whenever the integral is well deﬁned. This is the potential of g. Let PM be the class of functions deﬁned by PM = { G ∗ g : g ∈ EM }. For any function ϕ ∈ E ∗ , we deﬁne DM (ϕ) = inf u∈PM 1 2 R2 |∇ϕ − ∇u|2 dx (53) provided the distribution gradient of ϕ is locally square integrable, and we deﬁne DM (ϕ) to be inﬁnite otherwise.