Advances in Calculus of Variations
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Published By Walter De Gruyter Gmbh

1864-8266, 1864-8258

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Fares Essebei ◽  
Andrea Pinamonti ◽  
Simone Verzellesi

Abstract Given an open and bounded set Ω ⊆ ℝ n {\Omega\subseteq\mathbb{R}^{n}} and a family 𝐗 = ( X 1 , … , X m ) {\mathbf{X}=(X_{1},\ldots,X_{m})} of Lipschitz vector fields on Ω, with m ≤ n {m\leq n} , we characterize three classes of local functionals defined on first-order X-Sobolev spaces, which admit an integral representation in terms of X, i.e. F ⁢ ( u , A ) = ∫ A f ⁢ ( x , u ⁢ ( x ) , X ⁢ u ⁢ ( x ) ) ⁢ 𝑑 x , F(u,A)=\int_{A}f(x,u(x),Xu(x))\,dx, with f being a Carathéodory integrand.


2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Stefan Krömer ◽  
Martin Kružík ◽  
Elvira Zappale

Abstract For an integral functional defined on functions ( u , v ) ∈ W 1 , 1 × L 1 {(u,v)\in W^{1,1}\times L^{1}} featuring a prototypical strong interaction term between u and v, we calculate its relaxation in the space of functions with bounded variations and Radon measures. Interplay between measures and discontinuities brings various additional difficulties, and concentration effects in recovery sequences play a major role for the relaxed functional even if the limit measures are absolutely continuous with respect to the Lebesgue one.


2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Stefano Almi ◽  
Emanuele Tasso

Abstract We prove a compactness result in GBD {\operatorname{GBD}} which also provides a new proof of the compactness theorem in GSBD {\operatorname{GSBD}} , due to Chambolle and Crismale. Our proof is based on a Fréchet–Kolmogorov compactness criterion and does not rely on Korn or Poincaré–Korn inequalities.


2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Huyuan Chen ◽  
Laurent Véron

Abstract We provide bounds for the sequence of eigenvalues { λ i ⁢ ( Ω ) } i {\{\lambda_{i}(\Omega)\}_{i}} of the Dirichlet problem L Δ ⁢ u = λ ⁢ u ⁢  in  ⁢ Ω , u = 0 ⁢  in  ⁢ ℝ N ∖ Ω , L_{\Delta}u=\lambda u\text{ in }\Omega,\quad u=0\text{ in }\mathbb{R}^{N}% \setminus\Omega, where L Δ {L_{\Delta}} is the logarithmic Laplacian operator with Fourier transform symbol 2 ⁢ ln ⁡ | ζ | {2\ln\lvert\zeta\rvert} . The logarithmic Laplacian operator is not positively defined if the volume of the domain is large enough. In this article, we obtain the upper and lower bounds for the sum of the first k eigenvalues by extending the Li–Yau method and Kröger’s method, respectively. Moreover, we show the limit of the quotient of the sum of the first k eigenvalues by k ⁢ ln ⁡ k {k\ln k} is independent of the volume of the domain. Finally, we discuss the lower and upper bounds of the k-th principle eigenvalue, and the asymptotic behavior of the limit of eigenvalues.


2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Jan Knappmann ◽  
Henrik Schumacher ◽  
Daniel Steenebrügge ◽  
Heiko von der Mosel

Abstract We establish long-time existence for a projected Sobolev gradient flow of generalized integral Menger curvature in the Hilbert case and provide C 1 , 1 C^{1,1} -bounds in time for the solution that only depend on the initial curve. The self-avoidance property of integral Menger curvature guarantees that the knot class of the initial curve is preserved under the flow, and the projection ensures that each curve along the flow is parametrized with the same speed as the initial configuration. Finally, we describe how to simulate this flow numerically with substantially higher efficiency than in the corresponding numerical L 2 L^{2} gradient descent or other optimization methods.


2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Anthony Tromba

Abstract We prove for the first time that classical Morse theory applies to functionals of the form 𝒥 ⁢ ( u ) = 1 2 ⁢ ∫ Ω A α ⁢ β i ⁢ j ⁢ ( x ) ⁢ ∂ ⁡ u i ∂ ⁡ x α ⁢ ∂ ⁡ u j ∂ ⁡ x β ⁢ 𝑑 x + ∫ Ω G ⁢ ( x , u ) ⁢ 𝑑 x \displaystyle\mathcal{J}(u)=\frac{1}{2}\int_{\Omega}A^{ij}_{\alpha\beta}(x)% \frac{\partial u^{i}}{\partial x^{\alpha}}\frac{\partial u^{j}}{\partial x^{% \beta}}\,dx+\int_{\Omega}G(x,u)\,dx where u : Ω → ℝ N {u:\Omega\to\mathbb{R}^{N}} , Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} compact with C ∞ {C^{\infty}} boundary ∂ ⁡ Ω {\partial\Omega} , u | ∂ ⁡ Ω = φ {u|_{\partial\Omega}=\varphi} , and we argue that this is the largest class to which Morse theory applies.


2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Pierre Bousquet ◽  
Lorenzo Brasco ◽  
Chiara Leone ◽  
Anna Verde

Abstract We consider a quasilinear degenerate parabolic equation driven by the orthotropic p-Laplacian. We prove that local weak solutions are locally Lipschitz continuous in the spatial variable, uniformly in time.


2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Alessandro Carbotti ◽  
Simone Cito ◽  
Domenico Angelo La Manna ◽  
Diego Pallara

Abstract We prove the Γ-convergence of the renormalised Gaussian fractional s-perimeter to the Gaussian perimeter as s → 1 - {s\to 1^{-}} . Our definition of fractional perimeter comes from that of the fractional powers of Ornstein–Uhlenbeck operator given via Bochner subordination formula. As a typical feature of the Gaussian setting, the constant appearing in front of the Γ-limit does not depend on the dimension.


2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Danka Lučić ◽  
Tapio Rajala ◽  
Jyrki Takanen
Keyword(s):  

Abstract We prove an asymptotically sharp dimension upper-bound for the boundary of bounded simply-connected planar Sobolev W 1 , p {W^{1,p}} -extension domains via the weak mean porosity of the boundary. The sharpness of our estimate is shown by examples.


2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Davide Carazzato ◽  
Nicola Fusco ◽  
Aldo Pratelli

Abstract We consider functionals given by the sum of the perimeter and the double integral of some kernel g : ℝ N × ℝ N → ℝ + {g:\mathbb{R}^{N}\times\mathbb{R}^{N}\to\mathbb{R}^{+}} , multiplied by a “mass parameter” ε. We show that, whenever g is admissible, radial and decreasing, the unique minimizer of this functional among sets of given volume is the ball as soon as ε ≪ 1 {\varepsilon\ll 1} .


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