Minimax Theory Research Articles

Multiple critical points in closed sets via minimax theorems

In this paper, we apply our minimax theory ([Ricceri B. On a minimax theorem: an improvement, a new proof and an overview of its applications. Minimax Theory Appl. 2017;2:99–152; Ricceri B. A more complete version of a minimax theorem. Appl Anal Optim. 2021;5:251–261; Ricceri B. Addendum to “A more complete version of a minimax theorem”. Appl Anal Optim. 2022;6:195–197]) with the one developed by A. Moameni in [Moameni A. Critical point theory on convex subsets with applications in differential equations and analysis. J Math Pures Appl. 2020;141:266–315.] to formalize a general scheme giving the multiplicity of critical points. Here is a sample of application of the scheme to a critical elliptic problem: THEOREM. – Let Ω ⊂ R n ( n ≥ 3 ) be a smooth bounded domain and let 1 < q < 2 ≤ p < 2 n n − 2 . Then, for every r , ν > 0 , there exists λ ∗ > 0 with the following property: for every λ ∈ ] 0 , λ ∗ [ , μ ∈ ] − λ ∗ , λ ∗ [ , and for every convex dense set S ⊂ H − 1 ( Ω ) , there exists φ ~ ∈ S , with ‖ φ ~ ‖ H − 1 ( Ω ) < r , such that the problem { − Δu = λ ( | u | 4 n − 2 u + ν | u | q − 2 u + μ | u | p − 2 u + φ ~ ) in Ω u = 0 on ∂Ω has at least two solutions whose norms in H 0 1 ( Ω ) are less than or equal to r.

Boundary Behavior of Limit-Interfaces for the Allen–Cahn Equation on Riemannian Manifolds with Neumann Boundary Condition

We study the boundary behavior of any limit-interface arising from a sequence of general critical points of the Allen–Cahn energy functionals on a smooth bounded domain. Given any such sequence with uniform energy bounds, we prove that the limit-interface is a free boundary varifold which is integer rectifiable up to the boundary. This extends earlier work of Hutchinson and Tonegawa on the interior regularity of the limit-interface. A key novelty in our result is that no convexity assumption of the boundary is required and it is valid even when the limit-interface clusters near the boundary. Moreover, our arguments are local and thus work in the Riemannian setting. This work provides the first step towards the regularity theory for the Allen–Cahn min-max theory for free boundary minimal hypersurfaces, which was developed in the Almgren–Pitts setting by the first-named author and Zhou.

Min-max theory for capillary surfaces

Abstract We develop a min-max theory for the construction of capillary surfaces in 3-manifolds with smooth boundary. In particular, for a generic set of ambient metrics, we prove the existence of nontrivial, smooth, almost properly embedded surfaces with any given constant mean curvature 𝑐, and with smooth boundary contacting at any given constant angle 𝜃. Moreover, if 𝑐 is nonzero and 𝜃 is not π 2 \frac{\pi}{2} , then our min-max solution always has multiplicity one. We also establish a stable Bernstein theorem for minimal hypersurfaces with certain contact angles in higher dimensions.

Curvature Estimates for Stable Free Boundary Minimal Hypersurfaces in Locally Wedge-Shaped Manifolds

Abstract In this paper, we consider locally wedge-shaped manifolds, which are Riemannian manifolds that are allowed to have both boundary and certain types of edges. We define and study the properties of free boundary minimal hypersurfaces inside locally wedge-shaped manifolds. In particular, we show a compactness theorem for free boundary minimal hypersurfaces with curvature and area bounds in a locally wedge-shaped manifold. Additionally, using Schoen–Simon–Yau’s estimates, we also prove a Bernstein-type theorem indicating that, under certain conditions, a stable free boundary minimal hypersurface inside a Euclidean wedge must be a portion of a hyperplane. As our main application, we establish a curvature estimate for sufficiently regular free boundary minimal hypersurfaces in a locally wedge-shaped manifold with certain wedge angle assumptions. We expect this curvature estimate will be useful for establishing a min-max theory for the area functional in wedge-shaped spaces.

A variational proof of a disentanglement theorem for multilinear norm inequalities

The basic disentanglement theorem established by the present authors states that estimates on a weighted geometric mean over (convex) families of functions can be disentangled into quantitatively linked estimates on each family separately. On the one hand, the theorem gives a uniform approach to classical results including Maurey's factorisation theorem and Lozanovskiĭ's factorisation theorem, and, on the other hand, it underpins the duality theory for multilinear norm inequalities developed in our previous two papers.In this paper we give a simple proof of this basic disentanglement theorem. Whereas the approach of our previous paper was rather involved – it relied on the use of minimax theory together with weak*-compactness arguments in the space of finitely additive measures, and an application of the Yosida–Hewitt theory of such measures – the alternate approach of this paper is rather straightforward: it instead depends upon elementary perturbation and compactness arguments.

Multiplicity one for min–max theory in compact manifolds with boundary and its applications

Multiplicity one for min–max theory in compact manifolds with boundary and its applications

Formula omitted]output feedback fault-tolerant control of industrial processes based on zero-sum games and off-policy Q-learning

Traditional model-based control methods are often not applicable in industrial processes given the typical situation that model parameters are unknown, coefficient matrices are difficult to obtain, and system states are unpredictable. Accordingly, an output feedback fault-tolerant control method based on zero-sum game theory and off-policy Q-learning is presented in this study, with the aim of achieving smooth operation and good tracking performance for industrial processes that often contain sensor faults and disturbances. The specific steps are as follows. First, a system tracking error is introduced into the system to realize a novel extended model. Second, by establishing a performance index function and combining it with minimax theory, the fault-tolerant tracking control problem is converted into a zero-sum game problem. The Bellman and Riccati equations can be established after analyzing the relationship between the performance index and value functions. Then, the Q-function is introduced, and an off-policy Q-learning algorithm is combined with the Kronecker product without knowledge of system model parameters to design an optimal controller unbiased to detection noise. Finally, the effectiveness of the algorithm is verified by considering the injection molding process as an example. The experimental results validate that the designed controller demonstrates good control and extends the range of tolerable faults while maintaining good tracking performance.

Entire sign-changing solutions to the fractional [formula omitted]-Laplacian equation involving critical nonlinearity

We establish the existence of multiple sign-changing solutions to the fractional p-Laplacian equation (−△)psu−|u|p∗−2u=0,inRN,by combining the minimax theory and group equivariant technique with the concentration compactness argument.

Phase transitions for support recovery under local differential privacy

We address the problem of variable selection in a high-dimensional but sparse mean model, under the additional constraint that only privatized data are available for inference. The original data are vectors with independent entries having a symmetric, strongly log-concave distribution on \mathbb{R} . For this purpose, we adopt a recent generalization of classical minimax theory to the framework of local \alpha -differential privacy. We provide lower and upper bounds on the rate of convergence for the expected Hamming loss over classes of at most s -sparse vectors whose non-zero coordinates are separated from 0 by a constant a&gt;0 . As corollaries, we derive necessary and sufficient conditions (up to log factors) for exact recovery and for almost full recovery. When we restrict our attention to non-interactive mechanisms that act independently on each coordinate our lower bound shows that, contrary to the non-private setting, both exact and almost full recovery are impossible whatever the value of a in the high-dimensional regime such that n \alpha^2/ d^2\lesssim 1 . However, in the regime n\alpha^2/d^2\gg \log(d) we can exhibit a critical value a^* (up to a logarithmic factor) such that exact and almost full recovery are possible for all a\gg a^* and impossible for a\leq a^* . We show that these results can be improved when allowing for all non-interactive (that act globally on all coordinates) locally \alpha -differentially private mechanisms in the sense that phase transitions occur at lower levels.

Min-max theory for free boundary G-invariant minimal hypersurfaces

Given a compact Riemannian manifold Mn+1 with dimension 3≤n+1≤7 and ∂M≠∅, the free boundary min-max theory built by Li-Zhou [19] shows the existence of a smooth almost properly embedded minimal hypersurface with free boundary in M. In this paper, we generalize their constructions into equivariant settings. Specifically, let G be a compact Lie group acting as isometries on M with cohomogeneity at least 3. Then there exists a nontrivial smooth almost properly embedded free boundary G-invariant minimal hypersurface. Moreover, if the Ricci curvature of M is non-negative and ∂M is strictly convex, then there exist infinitely many properly embedded G-invariant minimal hypersurfaces with free boundary.

Existence of infinitely many minimal hypersurfaces in closed manifolds

Using min-max theory, we show that in any closed Riemannian manifold of dimension at least $3$ and at most $7$, there exist infinitely many smoothly embedded closed minimal hypersurfaces. It proves a conjecture of S.-T. Yau. This paper builds on the methods developed by F. C. Marques and A. Neves.

Research of the Fučik spectrum for the (p,q)-Laplacian operator by min-max theory

The object of research is the Fučik spectrum for the (p,q)-Laplacian operator. In the present paper, we are going to introduce the notion of the Fučik spectrum for a non-linear, non-homogeneous operator, which is the (p,q)-Laplacian operator through the study of the following eigenvalue boundary problem: {–∆pu–∆qu=λ(u+)p–1–μ(u–)q–1 in Ω, u=0 on ∂Ω, where Ω⊂RN, N≥1 is a bounded open subset with smooth boundary and λ and μ are two real parameters. In order to establish and show the existence of non-trivial solutions for the problem described above, we will search the weak solution of the energy functional associated to our problem by combining two essentials theorems of the Min-Max theory which are the Ljusternick-Schnirelmann (L-S) approach and the Col theorem. In addition to that, we are going to use the Ljusternick-Schnirelman theorem to show that our problem possesses a critical value ck in a suitable manifold that we will define later in the present manuscript. Following to that we will verify the Col geometry by using the critical point associated to the critical value ck and by applying the Col theorem, we will find a new critical value cn. After that, by employing the critical value cn we will demonstrate the existence of the family of curves which generate the set of Fučik spectrum of the (p,q)-Laplacian operator. To complete our research about the structure of the set of the Fučik spectrum of the (p,q)-Laplacian operator we will give the most important properties of the family of curves which are the continuity and the decrease. We have chosen to put our interest on the study of the Fučik spectrum because it’s determination is as important in mathematics as it is in many other fields (physics, plasma-physics, reaction-diffusion equation etc.). We can take as an example it’s use in the field of waves and vibrations where the starting point of the wave or the vibration is influenced by the structure and characteristics of the family of curves which constitute the Fučik spectrum of the (p,q)-Laplacian operator.

The first and second widths of the real projective space

In this paper, we deal with the first and second widths of the real projective space R P n \mathbb {RP}^{n} , for n n ranging from 4 4 to 7 7 , and for this we used some tools from the Almgren-Pitts min-max theory. In a recent paper, Ramirez-Luna computed the first width of the real projective spaces, and, at the same time, we obtained optimal sweepouts realizing the first and second widths of those spaces.

Generalized barycenters and variance maximization on metric spaces

We show that the variance of a probability measure \(\mu \) on a compact subset X of a complete metric space M is bounded by the square of the circumradius R of the canonical embedding of X into the space P(M) of probability measures on M, equipped with the Wasserstein metric. When barycenters of measures on X are unique (such as on CAT(0) spaces), our approach shows that R in fact coincides with the circumradius of X and so this result extends a recent result of Lim-McCann from Euclidean space. Our approach involves bi-linear minimax theory on \(P(X) \times P(M)\) and extends easily to the case when the variance is replaced by very general moments. As an application, we provide a simple proof of Jung’s theorem on CAT(k) spaces, a result originally due to Dekster and Lang-Schroeder.

A comparison of the Almgren–Pitts and the Allen–Cahn min–max theory

Min–max theory for the Allen–Cahn equation was developed by Guaraco (J Differ Geom 108:91–133, 2018) and Gaspar–Guaraco (Calc Var Partial Differ Equ 57:101, 42, 2018). They showed that the Allen–Cahn widths are greater than or equal to the Almgren–Pitts widths. In this article we will prove that the reverse inequalities also hold, i.e. the Allen–Cahn widths are less than or equal to the Almgren–Pitts widths. Hence, the Almgren–Pitts widths and the Allen–Cahn widths coincide. We will also show that all the closed minimal hypersurfaces (with optimal regularity), which are obtained from the Allen–Cahn min–max theory, are also produced by the Almgren–Pitts min–max theory. As a consequence, we will point out that the index upper bound in the Almgren–Pitts setting, proved by Marques–Neves (Camb J Math 4(4):463–511, 2016) and Li (An improved Morse index bound of min–max minimal hypersurfaces, 2020. arXiv:2007.14506 [math.DG]), can also be obtained from the index upper bound in the Allen–Cahn setting, proved by Gaspar (J Geom Anal 30:69-85, 2020) and Hiesmayr (Commun Partial Differ Equ 43(11):1541–1565, 2018).

Min–Max Theory for G-Invariant Minimal Hypersurfaces

In this paper, we consider a closed Riemannian manifold $$M^{n+1}$$ with dimension $$3\le n+1\le 7$$ , and a compact Lie group G acting as isometries on M with cohomogeneity at least 3. After adapting the Almgren–Pitts min–max theory to a G-equivariant version, we show the existence of a non-trivial closed smooth embedded G-invariant minimal hypersurface $$\Sigma \subset M$$ provided that the union of non-principal orbits forms a smooth embedded submanifold of M with dimension at most $$n-2$$ . Moreover, we also build upper bounds as well as lower bounds of (G, p)-widths, which are analogs of the classical conclusions derived by Gromov and Guth. An application of our results combined with the work of Marques–Neves shows the existence of infinitely many G-invariant minimal hypersurfaces when $$\mathrm{Ric}_M>0$$ and orbits satisfy the same assumption above.

Min–max minimal hypersurfaces with obstacle

We study min-max theory for area functional among hypersurfaces constrained in a smooth manifold with boundary. A Schoen-Simon-type regularity result is proved for integral varifolds which satisfy a variational inequality and restrict to a stable minimal hypersurface in the interior. Based on this, we show that for any admissible family of sweepouts $\Pi$ in a compact manifold with boundary, there always exists a closed $C^{1,1}$ hypersurface with codimension$\geq 7$ singular set in the interior and having mean curvature pointing outward along boundary realizing the width $\textbf{L}(\Pi)$.

Multilinear duality and factorisation for Brascamp–Lieb-type inequalities

We initiate the study of a duality theory which applies to norm inequalities for pointwise weighted geometric means of positive operators. The theory finds its expression in terms of certain pointwise factorisation properties of function spaces which are naturally associated to the norm inequality under consideration. We relate our theory to the Maurey–Nikishin–Stein theory of factorisation of operators, and present a fully multilinear version of Maurey’s fundamental theorem on factorisation of operators through L^1 . The development of the theory involves convex optimisation and minimax theory, functional-analytic considerations concerning the dual of L^\infty , and the Yosida–Hewitt theory of finitely additive measures. We consider the connections of the theory with the theory of interpolation of operators. We discuss the ramifications of the theory in the context of concrete families of geometric inequalities, including Loomis–Whitney inequalities, Brascamp–Lieb inequalities and multilinear Kakeya inequalities.

A Mountain-Pass Theorem for Asymptotically Conical Self-Expanders

We develop a min–max theory for asymptotically conical self-expanders of mean curvature flow. In particular, we show that given two distinct strictly stable self-expanders that are asymptotic to the same cone and bound a domain, there exists a new asymptotically conical self-expander trapped between the two.

CMC Doublings of Minimal Surfaces via Min–Max

Let \(\Sigma ^2 \subset M^3\) be a minimal surface of index 0 or 1. Assume that a neighborhood of \(\Sigma \) can be foliated by constant mean curvature (cmc) hypersurfaces. We use min–max theory and the catenoid estimate to construct \(\varepsilon \)-cmc doublings of \(\Sigma \) for small \(\varepsilon > 0\). Such cmc doublings were previously constructed for minimal hypersurfaces \(\Sigma ^n \subset M^{n+1}\) with \(n+1\ge 4\) by Pacard and Sun using gluing methods.