Applied Analysis and PDEs Seminar

October 4, 2024
Andreas Savas-Halilaj (University of Ioannina, Greece)
Sharp pinching theorems for complete CMC submanifolds in the sphere.
Abstract. A classical theorem due to Simons (Ann. of Math. 1968) and Lawson (Ann. Math. 1969) states that a compact, minimal n-dimensional minimal hypersurface in the sphere, whose second fundamental form satisfies |A| ≥ n, must be a either totally geodesic sphere or a Clifford torus. A similar result holds for compact hypersurfaces with non-zero constant mean curvature. In this talk, we show that that these results extend to complete immersed submanifolds. A significant obstacle to overcome in our case is that the maximum principles of Omori and Yau at infinity, which I will review during the talk, are not-applicable to our problem. The main tool is an adaptation of a conformal method developed by Fischer-Colbrie (Invent. Math. 1985). The results are contained in a joint work with M. Magliaro, L. Mari and F. Roing, (Crelle 2024).

October 11, 2024
Ioannis Giannoulis (University of Ioannina, Greece)
The semi-geostrophic equations and their relation to the Euler equations.
Abstract. The semi-geostrophic equations are two- or three-dimensional models that have been proposed in the 1970s as reduced models describing the large scale dynamics of weather fronts, based on the observation that these dynamics are determined mainly by the horizontal action of the Coriolis force arising from the rotation of the Earth around its axis. In their so called dual formulation, that is, after a suitable change of variables, the semi-geostrophic equations can be expressed as a coupled system of a scalar nonlinear transport equation with a velocity field that is determined through the solution of a Monge-Ampère equation which has the transported quantity as its right hand side. The existence of global in time weak solutions for this dual formulation has been proved by Benamou and Brenier in 1998. Later on, Loeper proved in 2006 the well-posedness of Hölder continuous solutions locally in time and showed that the dual semi-geostrophic equation in two dimensions can be seen as a perturbation of the Euler vorticity equation. In our talk we will focus on this latter relation and present results concerning its generalization and improved quantification, which have been obtained jointly with V. Kalivopoulos within a project funded by HFRI.

October 25, 2024
Dimitrios Gazoulis (University of Athens, Greece)
Mini courses on Homogenization Part I: Γ-convergence, G-convergence and their relation.
Abstract. We briefly introduce the notion of Γ-convergence and it's main properties together with two basic examples. The first example is the Γ-limit of the Modica-Mortola functional which relates phase transition type problems with minimal surfaces. The second, is an example from homogenization. Then, we introduce the notion of G-convergence for elliptic equations in divergence form and we state the homogenization theorem of G-convergence for elliptic operators. Finally, we illustrate the relationship between G-convergence with Γ-convergence and the second example stated in the beginning allow us to compare the G-limit with the weak* L^∞ limit.

November 1, 2024
Dimitrios Gazoulis (University of Athens, Greece)
Mini courses on Homogenization Part II: H-convergence and the Homogenization theorem.
Abstract. We introduce the notion of H-convergence for operators in divergence form that are not necessarily symmetric. We state the main properties of H-convergence and compare them with G-convergence. Then, we provide the main tool for the homogenization theorem based on compensated compactness, that is, the div-curl lemma. Finally, we state the homogenization theorem for H-convergence and define the corrector matrix that gives an approximation for the solutions of the homogenization problem, i.e. the corrector result.

November 8, 2024
Zhiyuan Geng (University of Purdue, US)
Uniqueness of the blow-down limit for the Allen-Cahn solution with a triple junction structure.
Abstract. In this talk, we investigate the vector-valued Allen-Cahn system with a triple-well potential, motivated by recent works of Alikakos-Geng and Sandier-Sternberg, which establish the existence of a minimizing entire solution exhibiting a triple junction structure at infinity along certain subsequences. We prove the uniqueness of the blow-down limit for this solution using a variational approach. Central to our analysis is a precise estimation of the diffuse interface’s location and size, achieved through tight upper and lower energy bounds. Furthermore, I will present new results on the asymptotic flatness of the diffuse interface at infinity, revealing that the solution is nearly invariant along the direction of the sharp interface.

November 22, 2024
Dimitrios Gazoulis (University of Athens, Greece)
Mini course on Homogenization Part III: The proof of the Homogenization theorem.
Abstract. We briefly recall the notion of H-convergence and the div-curl lemma that was proved in Part II of these series of lectures. Then, we provide an abstract setting in the spirit of Lax Milgram formulation that is utilized, together with the compensated compactness theorem, to complete the proof of the Homogenization theorem. Finally, we will prove the corrector result stated in the previous lecture, that gives an approximation for the solutions to the Homogenization problem.

November 29, 2024
Vassilis Rothos (Aristotle University of Thessaloniki , Greece)
Solitons in Discete NLS: Part I with saturable nonlinearity.
Abstract. We demonstrate existence of solitons in Discrete Nonlinear Schrodinger equation (DNLS) with saturable nonlinearity. We consider two types of solutions to DNLS periodic and vanishing at infinity. Calculus of variations and Nehari manifolds are employed to establish the existence of these solutions. We present some extensions of our results, combining the Nehari manifold approach and the Mountain Pass argument.

December 6, 2024
Orestis Vantzos (Department of Research, Technology and Development, IPTO, Greece)
Maximally Monotone Flows.
Abstract. We consider the notion of the flow of a maximally monotone operator A over a Hilbert space H, i.e. a solution of the initial-value problem u'(t) ∈ -A[u[t]], u(0)=u₀. The canonical example is the gradient flow of a (proper, convex, lower semicontinuous) functional I, where the maximally monotone operator is the subgradient of I. We discuss the well-posedness of such flows, and their time-discretization via minimal movements. Combined with a recently introduced class of splitting algorithms for determining the zeros of sums of maximally monotone operators, this technique opens up the numerical treatment of a class of interesting evolution problems.

December 6, 2024
Graham Patrick Benham (University College Dublin and the University of Oxford)
Getting a kick from water waves.
Abstract. Wave-driven propulsion occurs when a floating body, driven into oscillations at the fluid interface, is propelled by the waves generated by its own motion. In this seminar I will present a new theory for wave-driven propulsion based on coupling the equations of motion of a floating raft to a quasi-potential flow model of the fluid. Using this model, expressions are derived for the drift speed and propulsive thrust of the raft which in turn are shown to be consistent with global momentum conservation as well as experimental observations. I will also discuss some recent work using variational calculus to derive the optimum forcing for wave-driven propulsion, comparing analytical results with numerical optimization in the Julia programming language. References: Benham, G.P., Devauchelle, O., Morris, S.W. and Neufeld, J.A., 2022. Gunwale bobbing. Physical Review Fluids, 7(7), p.074804. https://arxiv.org/abs/2201.01533 Benham, G.P., Devauchelle, O. and Thomson, S.J., 2024. On wave-driven propulsion. Journal of Fluid Mechanics, 987, p.A44. https://arxiv.org/pdf/2310.12886