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# Interpolation inequalities in $\mathrm W^{1,p}( {\mathbb S}^1)$ and carré du champ methods

• * Corresponding author: Jean Dolbeault
• This paper is devoted to an extension of rigidity results for nonlinear differential equations, based on carré du champ methods, in the one-dimensional periodic case. The main result is an interpolation inequality with non-trivial explicit estimates of the constants in $\mathrm W^{1,p}( {\mathbb S}^1)$ with $p\ge2$. Mostly for numerical reasons, we relate our estimates with issues concerning periodic dynamical systems. Our interpolation inequalities have a dual formulation in terms of generalized spectral estimates of Keller-Lieb-Thirring type, where the differential operator is now a $p$-Laplacian type operator. It is remarkable that the carré du champ method adapts to such a nonlinear framework, but significant changes have to be done and, for instance, the underlying parabolic equation has a nonlocal term whenever $p\neq2$.

Mathematics Subject Classification: Primary: 35J92, 35K92; Secondary: 49K15, 58J35.

 Citation: • • Figure 1.  The vector field $(X,Y)\mapsto(|Y|^{\frac p{p-1}-2}\,Y,|X|^{p-2}\,X-|X|^{q-2}\,X)$ and periodic trajectories corresponding to $a = 1.35$ (with positive $X$) and $a = 1.8$ (with sign-changing $X$) are shown for $p = 2.5$ and $q = 3$. The zero-energy level is also shown

Figure 2.  The period $T_a$ of the solution of (12) with initial datum $X(0) = a\in(0,1)$ and $Y(0) = 0$ as a function of $a$ for $p = 3$ and $q = 5$. We observe that $\lim_{a\to0}T_a = +\infty$ and $\lim_{a\to1}T_a = 0$

Figure 3.  Left: the branch $\lambda\mapsto\mu(\lambda)$ for $p = 3$ and $q = 5$. Right: the curve $\lambda\mapsto\lambda-\mu(\lambda)$. In both cases, the bifurcation point $\lambda = \lambda_1^\star$ is shown by a vertical line

Figure 4.  The curves $p\mapsto\lambda_1$ (dotted) and $p\mapsto\lambda_1^\star$ (plain) differ

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