Absolutely minimising generalised solutions to the equations of vectorial calculus of variations in $$L^\infty $$

Consider the supremal functional

$$\begin{aligned} E_\infty (u,A) := \Vert \mathscr {L}(\cdot ,u,\mathrm {D}u)\Vert _{L^\infty (A)},\quad A\subseteq \Omega , \end{aligned}$$

(1)

applied to \(W^{1,\infty }\) maps \(u:\Omega \subseteq \mathbb {R}\longrightarrow \mathbb {R}^N\), \(N\ge 1\). Under certain assumptions on \(\mathscr {L}\), we prove for any given boundary data the existence of a map which is:

1. (i)
   a vectorial Absolute Minimiser of (1) in the sense of Aronsson,
    
2. (ii)
   a generalised solution to the ODE system associated to (1) as the analogue of the Euler-Lagrange equations,
    
3. (iii)
   a limit of minimisers of the respective \(L^p\) functionals as \(p\rightarrow \infty \) for any \(q\ge 1\) in the strong \(W^{1,q}\) topology and
    
4. (iv)
   partially \(C^2\) on \(\Omega \) off an exceptional compact nowhere dense set.
    

Our method is based on \(L^p\) approximations and stable a priori partial regularity estimates. For item ii) we utilise the recently proposed by the author notion of \(\mathcal {D}\)-solutions in order to characterise the limit as a generalised solution. Our results are motivated from and apply to Data Assimilation in Meteorology.

     

Korn’s inequality and John domains

Let \(\Omega \subset \mathbb {R}^n\), \(n\ge 2\), be a bounded domain satisfying the separation property. We show that the following conditions are equivalent:

1. (i)
   \(\Omega \) is a John domain;
    
2. (ii)
   for a fixed \(p\in (1,\infty )\), the Korn inequality holds for each \(\mathbf {u}\in W^{1,p}(\Omega ,\mathbb {R}^n)\) satisfying \(\int _\Omega \frac{\partial u_i}{\partial x_j}-\frac{\partial u_j}{\partial x_i}\,dx=0\), \(1\le i,j\le n\),

   $$\begin{aligned} \Vert D\mathbf {u}\Vert _{L^p(\Omega )}\le C_K(\Omega , p)\Vert \epsilon (\mathbf {u})\Vert _{L^p(\Omega )}; \qquad (K_{p}) \end{aligned}$$
    
3. (ii’)
   for all \(p\in (1,\infty )\), \((K_p)\) holds on \(\Omega \);
    
4. (iii)
   for a fixed \(p\in (1,\infty )\), for each \(f\in L^p(\Omega )\) with vanishing mean value on \(\Omega \), there exists a solution \(\mathbf {v}\in W^{1,p}_0(\Omega ,\mathbb {R}^n)\) to the equation \(\mathrm {div}\,\mathbf {v}=f\) with

   $$\begin{aligned} \Vert \mathbf {v}\Vert _{W^{1,p}(\Omega ,\mathbb {R}^n)}\le C(\Omega , p)\Vert f\Vert _{L^p(\Omega )};\qquad (DE_p) \end{aligned}$$
    
5. (iii’)
   for all \(p\in (1,\infty )\), \((DE_p)\) holds on \(\Omega \).
    

For domains satisfying the separation property, in particular, for finitely connected domains in the plane, our result provides a geometric characterization of the Korn inequality, and gives positive answers to a question raised by Costabel and Dauge (Arch Ration Mech Anal 217(3):873–898, 2015) and a question raised by Russ (Vietnam J Math 41:369–381, 2013). For the plane, our result is best possible in the sense that, there exist infinitely connected domains which are not John but support Korn’s inequality.

     

Relations between the $${\mathcal {}}$$-ultrafilters

Under CH we show the following results:

1. (1)
   There is a discrete ultrafilter which is not a \({\mathcal {Z}}_{0}\)-ultrafilter.
    
2. (2)
   There is a \(\sigma \)-compact ultrafilter which is not a \({\mathcal {Z}}_{0}\)-ultrafilter.
    
3. (3)
   There is a \({\mathcal {J}}_{\omega ^{3}}\)-ultrafilter which is not a \({\mathcal {Z}}_{0}\)-ultrafilter.
    

     

Derivations Vanishing Identities Involving Generalized Derivations and Multilinear Polynomial in Prime Rings

Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, \(f(x_1,\ldots ,x_n)\) be a multilinear polynomial over C, which is not central valued on R. Suppose that d is a non-zero derivation of R, F and G are two generalized derivations of R such that \(d\{F(u)u-uG^2(u)\}=0\) for all \(u\in f(R)\). Then one of the following holds:

1. (i)
   there exist \(a, b, p\in U\), \(\lambda \in C\) such that \(F(x)=\lambda x+bx+xa^2\), \(G(x)=ax\), \(d(x)=[p, x]\) for all \(x\in R\) with \([p, b]=0\) and \(f(x_1,\ldots , x_n)^2\) is central valued on R;
    
2. (ii)
   there exist \(a, b, p\in U\) such that \(F(x)=ax\), \(G(x)=xb\), \(d(x)=[p,x]\) for all \(x\in R\) and \(f(x_1,\ldots , x_n)^2\) is central valued on R with \([p, a-b^2]=0\);
    
3. (iii)
   there exist \(a\in U\) such that \(F(x)=xa^2\) and \(G(x)=ax\) for all \(x\in R\);
    
4. (iv)
   there exists \(a\in U\) such that \(F(x)=a^2x\) and \(G(x)=xa\) for all \(x\in R\) with \(a^2\in C\);
    
5. (v)
   there exist \(a, p\in U\), \(\lambda , \alpha , \mu \in C\) such that \(F(x)=\lambda x-a^2x\), \(G(x)=xa\) and \(d(x)=[p,x]\) for all \(x\in R\) with \(a^2=\mu -\alpha p\) and \(\alpha p^2+(\lambda -2\mu ) p\in C\);
    
6. (vi)
   there exist \(a\in U\), \(\lambda \in C\) such that R satisfies \(s_4\) and either \(F(x)=\lambda x+xa^2\), \(G(x)=ax\) or \(F(x)=\lambda x-a^2x\), \(G(x)=xa\) for all \(x\in R\).
    

     

Mean dimension of $${\mathbb{Z}^k}$$-actions

Mean dimension is a topological invariant for dynamical systems that is meaningful for systems with infinite dimension and infinite entropy. Given a \({\mathbb{Z}^k}\)-action on a compact metric space X, we study the following three problems closely related to mean dimension.

1. (1)
   When is X isomorphic to the inverse limit of finite entropy systems?
    
2. (2)
   Suppose the topological entropy \({h_{\rm top}(X)}\) is infinite. How much topological entropy can be detected if one considers X only up to a given level of accuracy? How fast does this amount of entropy grow as the level of resolution becomes finer and finer?
    
3. (3)
   When can we embed X into the \({\mathbb{Z}^k}\)-shift on the infinite dimensional cube \({([0,1]^D)^{\mathbb{Z}^k}}\)?
    

These were investigated for \({\mathbb{Z}}\)-actions in Lindenstrauss (Inst Hautes Études Sci Publ Math 89:227–262, 1999), but the generalization to \({\mathbb{Z}^k}\) remained an open problem. When X has the marker property, in particular when X has a completely aperiodic minimal factor, we completely solve (1) and a natural interpretation of (2), and give a reasonably satisfactory answer to (3).A key ingredient is a new method to continuously partition every orbit into good pieces.     

On a <Emphasis Type="Italic">K</Emphasis>-component elliptic system with the Sobolev critical exponent in high dimensions: the repulsive case

Study the following K-component elliptic system

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Here \(k\ge 2\) is a integer and \(\Omega \subset \mathbb {R}^N(N\ge 4)\) is a bounded domain with smooth boundary \(\partial \Omega \), \(a_i,\lambda _i>0\), \(b_i\ge 0\) for all \(i=1,2,\ldots ,k\) and \(\beta <0\), \(2^*=\frac{2N}{N-2}\) is the Sobolev critical exponent. By the variational method, we obtain a nontrivial solution of this system. The concentration behavior of this nontrivial solution as \(\overrightarrow{\mathbf {b}}\rightarrow \overrightarrow{\mathbf {0}}\) and \(\beta \rightarrow -\infty \) are both studied and the phase separation is exhibited for \(N\ge 6\), where \(\overrightarrow{\mathbf {b}}=(b_1,b_2,\ldots ,b_k)\) is a vector. Our results extend and generalize the results in Chen and Zou  (Arch Ration Mech Anal 205:515–551, 2012; Calc Var Partial Differ Equ 52:423–467, 2015). Moreover, by studying the phase separation, we also prove some existence and multiplicity results of the sign-changing solutions to the following Brezís–Nirenberg problem of the Kirchhoff type

$$\begin{aligned} \left\{ \begin{array}{ll} -\bigg (a+b\int _{\Omega }|\nabla u|^2dx\bigg )\Delta u = \lambda u +|u|^{2^*-2}u, &{}\quad \text {in }\Omega , \\ u =0,&{}\quad \text {on }\partial \Omega , \end{array} \right. \end{aligned}$$

where \(N\ge 6\), \(a,\lambda >0\) and \(b\ge 0\). These results can be seen as an extension of the results in Cerami et al. (J Funct Anal 69:289–306, 1986). The concentration behaviors of the sign-changing solutions to the above equation as \(b\rightarrow 0^+\) are also obtained.

     

Semistability and Restrictions of tangent bundle to curves

We consider all complex projective manifolds X that satisfy at least one of the following three conditions:

1. (1)
   There exists a pair \({(C\,,\varphi)}\) , where C is a compact connected Riemann surface and

   $\varphi\,:\, C\,\longrightarrow\, X$

   a holomorphic map, such that the pull back \({\varphi^* {\it TX}}\) is not semistable.
    
2. (2)
   The variety X admits an étale covering by an abelian variety.
    
3. (3)
   The dimension dim X ≤ 1.
    

We prove that the following classes are among those that are of the above type.

• All X with a finite fundamental group.
• All X such that there is a nonconstant morphism from \({{\mathbb C}{\mathbb P}^1}\) to X.
• All X such that the canonical line bundle K _X is either positive or negative or \({c_1(K_X)\,\in\,H^2(X,\, {\mathbb Q})}\) vanishes.
• All X with \({{\rm dim}_{\mathbb C} X\, =\,2}\).

     

Classification of minimal mass blow-up solutions for an $${L^{2}}$$ critical inhomogeneous NLS

We establish the classification of minimal mass blow-up solutions of the \({L^{2}}\) critical inhomogeneous nonlinear Schrödinger equation

$$i\partial_t u + \Delta u + |x|^{-b}|u|^{\frac{4-2b}{N}}u = 0,$$

thereby extending the celebrated result of Merle (Duke Math J 69(2):427–454, 1993) from the classic case \({b=0}\) to the case \({0< b< {\rm min} \{2,N\} }\), in any dimension \({N \geqslant 1}\).

     

A semilinear parabolic problem with singular term at the boundary

In this paper, we deal with a class of semilinear parabolic problems related to a Hardy inequality with singular weight at the boundary.

More precisely, we consider the problem

$$\left\{\begin{array}{l@{\quad}l}u_t-\Delta u=\lambda \frac{u^p}{d^2}&\text{ in }\,\Omega_{T}\equiv\Omega \times (0,T), \\u>0 &\text{ in }\,{\Omega_T}, \\u(x,0)=u_0(x)>0 &\text{ in }\,\Omega, \\u=0 &\text{ on }\partial \Omega \times (0,T),\end{array}\right.$$

(P)

where Ω is a bounded regular domain of \({\mathbbm{R}^N}\), \({d(x)=\text{dist}(x,\partial\Omega)}\), \({p > 0}\), and \({\lambda > 0}\) is a positive constant.

We prove that

1. 1.
   If \({0 < p < 1}\), then (P) has no positive very weak solution.
    
2. 2.
   If \({p=1}\), then (P) has a positive very weak solution under additional hypotheses on \({\lambda}\) and \({u_0}\).
    
3. 3.
   If \({p > 1}\), then, for all \({\lambda > 0}\), the problem (P) has a positive very weak solution under suitable hypothesis on \({u_0}\).
    

Moreover, we consider also the concave–convex-related case.

     

On the numerical approximation of $$\infty $$-harmonic mappings

A map \(u : \Omega \subseteq \mathbb {R}^n \longrightarrow \mathbb {R}^N\), is said to be \(\infty \)-harmonic if it satisfies

Open image in new window

(1)

The system (1) is the model of vector-valued Calculus of Variations in \(L^\infty \) and arises as the “Euler-Lagrange” equation in relation to the supremal functional

$$\begin{aligned} E_\infty (u,\Omega )\, :=\, \Vert \text {D}u \Vert _{L^\infty (\Omega )}. \end{aligned}$$

(2)

In this work we provide numerical approximations of solutions to the Dirichlet problem when \(n=2\) and in the vector valued case of \(N=2,3\) for certain carefully selected boundary data on the unit square. Our experiments demonstrate interesting and unexpected phenomena occurring in the vector valued case and provide insights on the structure of general solutions and the natural separation to phases they present.

     

Zero varieties for the Nevanlinna class in weakly pseudoconvex domains of maximal type F in $$\mathbb {C}^2$$

Let \(\Omega \) be a bounded, uniformly totally pseudoconvex domain in \(\mathbb {C}^2\) with smooth boundary \(b\Omega \). Assume that \(\Omega \) is a domain admitting a maximal type F. Here, the condition maximal type F generalizes the condition of finite type in the sense of Range (Pac J Math 78(1):173–189, 1978; Scoula Norm Sup Pisa, pp 247–267, 1978) and includes many cases of infinite type. Let \(\alpha \) be a d-closed (1, 1)-form in \(\Omega \). We study the Poincaré–Lelong equation

$$\begin{aligned} i\partial \bar{\partial }u=\alpha \quad \text {on}\, \Omega \end{aligned}$$

in \(L^1(b\Omega )\) norm by applying the \(L^1(b\Omega )\) estimates for \(\bar{\partial }_b\)-equations in [11]. Then, we also obtain a prescribing zero set of Nevanlinna holomorphic functions in \(\Omega \).

     

The Riesz transform and quantitative rectifiability for general Radon measures

In this paper we show that if \(\mu \) is a Borel measure in \({{\mathbb {R}}}^{n+1}\) with growth of order n, such that the n-dimensional Riesz transform \({{\mathcal {R}}}_\mu \) is bounded in \(L^2(\mu )\), and \(B\subset {{\mathbb {R}}}^{n+1}\) is a ball with \(\mu (B)\approx r(B)^n\) such that:

1. (a)
   there is some n-plane L passing through the center of B such that for some \(\delta >0\) small enough, it holds

   $$\begin{aligned}\int _B \frac{\mathrm{dist}(x,L)}{r(B)}\,d\mu (x)\le \delta \,\mu (B),\end{aligned}$$
    
2. (b)
   for some constant \({\varepsilon }>0\) small enough,

   $$\begin{aligned}\int _{B} |{{\mathcal {R}}}_\mu 1(x) - m_{\mu ,B}({{\mathcal {R}}}_\mu 1)|^2\,d\mu (x) \le {\varepsilon }\,\mu (B),\end{aligned}$$

   where \(m_{\mu ,B}({{\mathcal {R}}}_\mu 1)\) stands for the mean of \({{\mathcal {R}}}_\mu 1\) on B with respect to \(\mu \),
    

then there exists a uniformly n-rectifiable set \(\Gamma \), with \(\mu (\Gamma \cap B)\gtrsim \mu (B)\), and such that \(\mu |_\Gamma \) is absolutely continuous with respect to \({{\mathcal {H}}}^n|_\Gamma \). This result is an essential tool to solve an old question on a two phase problem for harmonic measure in subsequent papers by Azzam, Mourgoglou, Tolsa, and Volberg.

     

Anonymous and leakage resilient IBE and IPE

We construct identity-based encryption and inner product encryption schemes under the decision linear assumption. Their private user keys are leakage-resilient in several scenarios. In particular,

• In the bounded memory leakage model (Akavia et al., TCC, vol. 5444, pp. 474–495, 2009), our basic schemes reach the maximum-possible leakage rate \(1-o(1)\).
• In the continual memory leakage model (Brakerski et al., Overcoming the hole in the bucket: public-key cryptography resilient to continual memory leakage, 2010; Dodis et al., Cryptography against continuous memory attacks, 2010), variants of the above schemes enjoy leakage rate at least \(\frac{1}{2} -o(1)\). Among the results, we improve upon the work of Brakerski et al. by presenting adaptively secure IBE schemes.

In addition, we prove that our IBE schemes are anonymous under the DLIN assumption, so that ciphertexts leaks no information on the corresponding identities. Similarly, attributes in IPE are proved computationally hidden in the corresponding ciphertexts.

     

Chains of saturated models in AECs

We study when a union of saturated models is saturated in the framework of tame abstract elementary classes (AECs) with amalgamation. We prove:

Theorem 0.1. If K is a tame AEC with amalgamation satisfying a natural definition of superstability (which follows from categoricity in a high-enough cardinal), then for all high-enough \(\lambda {:}\)

1. (1)
   The union of an increasing chain of \(\lambda \)-saturated models is \(\lambda \)-saturated.
    
2. (2)
   There exists a type-full good \(\lambda \) -frame with underlying class the saturated models of size \(\lambda \).
    
3. (3)
   There exists a unique limit model of size \(\lambda \).
    

Our proofs use independence calculus and a generalization of averages to this non first-order context.

     

Chvátal–Erdös Type Conditions for Hamiltonicity of Claw-Free Graphs

For a graph H, let \(\alpha (H)\) and \(\alpha ^{\prime }(H)\) denote the independence number and the matching number, respectively. Let \(k\ge 2\) and \(r>0\) be given integers. We prove that if H is a k-connected claw-free graph with \(\alpha (H)\le r\), then either H is Hamiltonian or the Ryjá c? ek’s closure \(cl(H)=L(G)\) where G can be contracted to a k-edge-connected \(K_3\)-free graph \(G_0^{\prime }\) with \(\alpha ^{\prime }(G_0^{\prime })\le r\) and \(|V(G_0^{\prime })|\le \max \{3r-5, 2r+1\}\) if \(k\ge 3\) or \(|V(G_0^{\prime })|\le \max \{4r-5, 2r+1\}\) if \(k=2\) and \(G_0^{\prime }\) does not have a dominating closed trail containing all the vertices that are obtained by contracting nontrivial subgraphs. As corollaries, we prove the following:

1. (a)
   A 2-connected claw-free graph H with \(\alpha (H)\le 3\) is either Hamiltonian or \(cl(H)=L(G)\) where G is obtained from \(K_{2,3}\) by adding at least one pendant edge on each degree 2 vertex;
    
2. (b)
   A 3-connected claw-free graph H with \(\alpha (H)\le 7\) is either Hamiltonian or \(cl(H)=L(G)\) where G is a graph with \(\alpha ^{\prime }(G)=7\) that is obtained from the Petersen graph P by adding some pendant edges or subdividing some edges of P.
    

Case (a) was first proved by Xu et al. [19]. Case (b) is an improvement of a result proved by Flandrin and Li [12]. For a given integer \(r>0\), the number of graphs of order at most \(\max \{4r-5, 2r+1\}\) is fixed. The main result implies that improvements to case (a) or (b) by increasing the value of r and by enlarging the collection of exceptional graphs can be obtained with the help of a computer. Similar results involved degree or neighborhood conditions are also discussed.