Schedule for: 24w5503 - Nonlocal Problems in Mathematical Physics, Analysis and Geometry

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

A brief introduction with important logistical information, technology instruction, and opportunity for participants to ask questions.

I will report some recent results on the nonlocal Fisher-KPP equation with a free boundary, which is regarded as a model for the propagation of a new or invasive species, with the free boundary representing the spreading front. We will pay particular attention to the rate of propagation and give precise estimate for such rate, including the case of accelerated propagation, a phenomenon that can only appear in the Fisher-KPP model when the diffusion is nonlocal.

Boundary Hardy inequality states that if $1 < p < \infty$ and $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^d$, then \begin{equation}\label{clhar} \int_{\Omega} \frac{|u(x)|^{p}}{\delta^{p}_{\Omega}(x)} dx \leq C\int_{\Omega} |\nabla u(x) |^{p}dx, \text{ for all } \ u \in C^{\infty}_{c}(\Omega), \end{equation} where $\delta_\Omega(x)$ is the distance function from $\partial\Omega$. B. Dyda generalised the above inequality to the fractional setting, which says, for $sp >1$ and $s\in (0,1)$ \begin{equation}\label{fractionalhardy} \int_{\Omega} \frac{|u(x)|^{p}}{\delta_{\Omega}^{sp}(x)} dx \leq C \int_{\Omega} \int_{\Omega} \frac{|u(x)-u(y)|^{p}}{|x-y|^{d+sp}} dxdy, \text{ for all } \ u \in C^{\infty}_{c}(\Omega). \end{equation} \eqref{clhar} and \eqref{fractionalhardy} is not true for $p=1$ and $sp=1$ respectively. In this talk, I will present the appropriate inequalities for the critical cases: $p=1$ for \eqref{clhar} and $sp= 1$ for \eqref{fractionalhardy}. If time permits, I will discuss the case when the weight function ($\delta_\Omega$) in \eqref{fractionalhardy} is replaced by distance function from a $k-$ dimensional sub manifold of $\Omega$.

In this talk, we will consider blow-up criterion for the well-known fully parabolic Keller-Segel model with in between two critical exponents. The objective is to determine the global existence and finite time blow-up of weak solutions by utilizing a combination of invariant norms of the initial data. The results are consistent with the parabolic-elliptic version of chemotaxis system.

Let $f\in L^\infty(\Omega)$ and $u_f$ be the unique solution of the equation $$(-\Delta)^s u =f$$ in $\Omega$, $u=0$ in $\Omega^c$. Here $(-\Delta)^s$ is the fractional Laplacian and the solution is defined in the weak sense $$ \int_\Omega fvdx=c_s \iint_{{\mathbb R}^{2n}} \frac{(u_f(x)-u_f(y)(v(x)-v(y))}{|x-y|^{n+2s}}dxdy, $$ for all $v\in C_0^\infty(\Omega)$.

Problem 1: Consider the energy functional $$ \Phi_s(f)= c_s \iint_{{\mathbb R}^{2n}} \frac{(u_f(x)-u_f(y))^2}{|x-y|^{n+2s}}dxdy, $$ and minimize $\Phi_s$ over the convex closed set $$ \bar{{\mathcal R}}_\beta=\left\{f\in L^\infty(D)\colon 0\leq f \leq 1,\,\,\int_D fdx=\beta \right\}. $$ We show the existence of the unique minimizer $f$. Moreover, for $\alpha=\max u_f$ the function $u=\alpha-u_{f}$ solves $$ -(-\Delta)^s u-\chi_{\{u\leq 0\}}\min\{-(-\Delta)^s u^+;1\}=\chi_{\{u>0\}}. $$ Problem 2: In the case $s=1$ we obtain the classical rearrangement problem for the Laplacian with the energy $\Phi(f)=\|\nabla u_f\|_2^2$. Here we minimize $\Phi(f)$ over the rearrangement set with constraint in the cylindrical domain $\Omega=D\times (0,1)$ $$ {\bar{\mathcal R}}^D_\beta=\left\{f(x)\in L^\infty(\Omega)\colon f(x',x_n)=f(x'),\,\, 0\leq f \leq 1,\,\,\int_D fdx=\beta \right\}. $$ We show the existence of the unique minimizer. Moreover, we show that for a particular $\alpha>0$ the function $U=\alpha-u_f$ minimizes the functional with with nonlocal obstacle acting on function $V(x')=\int_0^1 U(x', t) dt $ $$ \int_\Omega \frac{1}{2}|\nabla U(x)|^2dx +\int_D V(x')^+\,dx', $$ and solves the equation $$ \Delta U(x',x_n) = \chi_{\{V>0\}}(x') + \chi_{\{V=0\}}(x') [\partial_\nu U (x',0) + \partial_\nu U (x',1)], $$ where $\partial_\nu U$ is the exterior normal derivative of $U$. Several further regularity results are proven. It is shown that the comparison principle does not hold for minimizers, which makes numerical approximation we present somehow challenging.

Lunch is served daily between 11:30am and 1:30pm in the Xianghu Lake National Tourist Resort

In this talk I will discuss a boundary value problem for the p-Laplacian, posed in the exterior of a large number of small cavities that all have the same p-capacity. These cavities are anchored to the unit sphere in ${\mathbb R}^d$, where 1$<$p$<$d and the anchoring points are asymptotically uniformly distributed. The solution is required to be 1 on all cavities and decay to 0 at infinity. I will discuss qualitative properties of the solution for this problem in different parametric regimes. In particular, it will be shown that the solution of this problem can be approximated in $L^\infty$ by a surprisingly simple ansatz function which can be explicitly computed in terms of the parameters of the problem. This talk is based on a joint work with Fedor Nazarov and Yuval Peres.

We obtain some Trudinger-Moser inequalities in the plane unit disc where the Leray potential is involved. We provide also some improved Leray inequality with different types of reminder terms. This is based on joint work with HY Chen and Yihong Du.

The goal of this talk is to explore the asymptotic behaviour of anisotropic pro\-blems governed by operators of the pseudo $p$-Laplacian type i.e. \begin{equation*} \begin{cases} - \sum_i \partial_{x_i}\Big(|\partial_{x_i} u_\ell |^{pi-2}\partial_{x_i} u_\ell \Big)= f ~~\text{in} ~\Omega_\ell=\ell \omega_1 \times \omega_2, \\ u_\ell = 0 ~~\text{on} ~\partial\Omega_\ell, \end{cases} \end{equation*} when the size $\ell$ of the domain $\Omega_\ell$ goes to infinity.

Reference

M. Chipot, Asymptotic behaviour of some anisotropic problems, Asymptotic Analysis, to appear.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

In this talk, the asymptotic behavior of a semilinear heat equation with long time memory and non-local diffusion is analyzed in the usual set-up for dynamical systems generated by differential equations with delay terms. This approach is different from the previous published literature on the long time behavior of heat equations with memory which is carried out by the Dafermos transformation. As a consequence, the obtained results provide complete information about the attracting sets for the original problem, instead of the transformed one. In particular, the proved results also generalize and complete previous literature in the local case (see [1],[2]).

References:

[1] J. Xu, T. Caraballo, J. Valero, Asymptotic behavior of a semilinear problem in heat conduction with long time memory and non-local diffusion, Journal of Differential Equations 327 (2022), 418-447.

[2] J. Xu, T. Caraballo, J. Valero, Asymptotic behavior of nonlocal partial differential equations with long time memory. Discrete and Continuous Dynamical Systems, Series S 15 (2022), no. 10, 3059-3078.

In this talk, I will introduce some new results about the existence and nonexistence of principal eigenvalues of nonlocal diffusion operators with sign-changing advection on a circle.

We consider maps between spheres $S^n$ to $S^\ell$ that minimize the Sobolev-space energy $W^{s,n/s}$ for some $s \in (0,1)$ in a given homotopy class. The basic question is: in which homotopy class does a minimizer exist? This is a nontrivial question since the energy under consideration is conformally invariant and bubbles can form. Sacks-Uhlenbeck theory tells us that minimizers exist in a set of homotopy classes that generates the whole homotopy group $\pi_{n}(S^\ell)$. In some situations explicit examples are known if $n/s = 2$ or $s=1$. In our talk we are interested in the stability of the above question in dependence of s. We can show that as s varies locally, the set of homotopy classes in which minimizers exist can be chosen stable. We also discuss that the minimum $W^{s,n/s}$-energy in homotopy classes is continuously depending on $s$. Joint work with K. Mazowiecka (U Warsaw)

We study the minimization of the energy functional associated with cholesteric liquid crystals. We consider two types of problems: 1. Stability of a stationary solution where the molecules rotate at an “unnatural” rate (and which cannot be, therefore, the global minimizer of the associated energy functional). 2. Assuming one-dimensional behavior, we obtain the global minimizer for the same boundary conditions as in the first problem.

This talk concerns the Ginzburg-Landau equation and the Meissner equation on a 3-dimensional multiply-connected domain. The novelty of these equations is that each of them contains an unknown potential and an unknown Neumann field as the Lagrange multipliers, which indicates the connection between the Ginzburg-Landau equation and the Maxwell-Stokes system, hence suggests the similarity of the effects of domain topology on these equations. In this talk we shall show new results of existence of the minimal and non-minimal solutions for each equation, which emphasize the effect of the domain topology. The various estimates of the Maxwell-Stokes system play an important role in our approach.

Aggregation-diffusion equations are involved in modelling many phenomena, in particular in astrophysics and in biology (chemotaxis). The most well-known example is the one of the Keller- Segel (KS) system. Here we consider a class of KS-type models with solutions which explose in the zero-diffusion limit. We characterise in a sharp way their behaviour (concentration, Lebesgue norms) in the small-diffusion regime in the radially symmetric case. We will compare our results with previous analogous ones for scalar conservation laws. The research has been done in collaboration with P. Biler, G. Karch (Wroclaw) and P. Laurençot (Toulouse)

The talk is on the full range of Caffarelli-Kohn-Nirenberg’s inequalities associated with the Coulomb terms when the derivative is between 0 and 1. The radial case is also discussed. This is joint work with Arka Mallick.

In the last years, elliptic equations involving a nonsmooth term have attracted several outstanding mathematicians and the interest towards this kind of problems has grown more and more, not only for their intriguing analytical structure, but also in view of their applications in a wide range of contexts. Motivated by this wide interest in the literature, the leading purpose of this talk is to present some recent results on nonsmooth elliptic equations, mainly related to a wide class of functionals defined through multiple integrals of Calculus of Variations. Applications to quasilinear boundary value problems will be presented and some open problems briefly discussed; see [1] and {2, Chapter 8] for related topics.

[1] C. Alves, G. Molica Bisci, and S. da Silva, New minimax theorems for lower semicontinuous functions and applications}, ESAIM: Control, Optimisation and Calculus of Variations. DOI: https://doi.org/10.1051/cocv/2024005 (in press).

[2] G. Molica Bisci and P. Pucci, Nonlinear Problems with Lack of Compactness, De Gruyter Series in Nonlinear Analysis and Applications {36} (2021), i+vii, 1--266.

In this talk we deal with some nonlocal problem driven by the fractional Laplacian in presence of jumping nonlinearities. Using variational and topological methods, we prove the existence of a nontrivial solution for the problem under consideration. These existence results can be seen as the nonlocal counterpart of the ones obtained in the context of the Laplacian equations. In the nonlocal framework the arguments used in the classical setting have to be refined. Indeed the presence of the fractional Laplacian operator gives rise to some additional difficulties, that we are able to overcome proving new regularity results for weak solutions of nonlocal problems, which are of independent interest. This is a joint paper with Giovanni Molica Bisci, Kanishka Perera and Caterina Sportelli (Journal des Mathématiques Pures et Appliquées, 2024).

I will present recent results on modeling of ferroelectric nematics using a Ginzburg-Landau-type model with anisotropic elastic constants. We show that the singular structures exhibited by the minimizers of this energy contain both point and line singularities and describe these structures for particular examples of domains/boundary conditions. This is a joint work with Priyanka Kumari, Oleg Lavrentovich and Peter Sternberg.

We mainly consider a priori estimates of possibly sign-changing global solutions to superlinear parabolic problems and their applications (blow-up rates, existence of nontrivial steady states etc.). Our estimates are based on energy, interpolation and bootstrap arguments. We first discuss some known results on local problems and then consider problems with nonlocal nonlinearities and/or nonlocal differential operators.

We analyze functions in Besov spaces $B^{1/q}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d),q\in (1,\infty)$, and functions in fractional Sobolev spaces $W^{r,q}(\mathbb{R}^N,\mathbb{R}^d),r\in (0,1),q\in [1,\infty)$. We prove for Besov functions $u\in B^{1/q}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d)$ the summability of the difference between one-sided approximate limits in power $q$, $|u^+-u^-|^q$, along the jump set $\mathcal{J}_u$ of $u$ with respect to Hausdorff measure $\mathcal{H}^{N-1}$, and establish the best bound from above on the integral $\int_{\mathcal{J}_u}|u^+-u^-|^qd\mathcal{H}^{N-1}$ in terms of Besov constants. We show for functions $u\in B^{1/q}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d),q\in (1,\infty)$ that \begin{equation*} \liminf\limits_{\varepsilon\to 0^+}\frac{1}{\varepsilon^N}\int_{B_{\varepsilon}(x)} |u(z)-u_{B_{\varepsilon}(x)}|^qdz=0 \end{equation*} for every $x$ outside of a $\mathcal{H}^{N-1}$-sigma finite set. For Sobolev functions $u\in W^{r,q}(\mathbb{R}^N,\mathbb{R}^d)$ we prove that \begin{equation*} \lim\limits_{\varepsilon\to 0^+}\frac{1}{\varepsilon^N}\int_{B_{\varepsilon}(x)} |u(z)-u_{B_{\varepsilon}(x)}|^qdz=0 \end{equation*} for $\mathcal{H}^{N-rq}$ a.e. $x$, where $q\in[1,\infty)$, $r\in(0,1]$ and $rq\leq N$. In addition, we prove Lusin-type approximation for fractional Sobolev functions $u\in W^{r,q}(\mathbb{R}^N,\mathbb{R}^d)$ by H{\"o}lder continuous functions in $C^{0,r}(\mathbb{R}^N,\mathbb{R}^d)$.

I will introduce a new model of liquid crystal flows coupling the harmonic map heat flow with free boundary with a Navier-Stokes system. I will prove existence in 2D of partially smooth solutions and provide an example of solution blowing-up in finite time. I will also address some open problems.

We consider a thermodynamically consistent diffuse interface model for viscous incompressible two-phase flows where the mechanisms of chemotaxis, active transport and nonlocal interaction of Oono's type are taken into account. The evolution system couples the Navier-Stokes equations for the volume-averaged fluid velocity, a convective Cahn-Hilliard equation for the phase-field variable, and an advection-diffusion equation for the density of a chemical substance. For the initial-boundary value problem with a physically relevant singular potential in three dimensions, we report some recent results on its well-posedness and long-time behavior of global solutions.

In this talk, I will present a Schauder-type estimate for general local and non-local linear parabolic system $$\partial_tu+\mathcal{L}_su=\Lambda^\gamma f+g$$ in $(0,\infty)\times\mathbb{R}^d$ where $\Lambda=(-\Delta)^{\frac{1}{2}}$, $0<\gamma\leq s$, $\mathcal{L}_s$ is the Pesudo-differential operator of the order s. By applying our Schauder-type estimate to suitably chosen differential operators $\mathcal{L}_s$, we obtain critical well-posedness results of various local and non-local nonlinear evolution equations in geometry and fluids, including hypoviscous Navier--Stokes equations, the surface quasi-geostrophic equation, mean curvature equations, Willmore flow, surface diffusion flow, Peskin equations, thin-film equations and Muskat equations.

The Choquard equation is a partial differential equation that hasgained significant interest and attention in recent decades. It is anonlinear equation that combines elements of both the Laplace andSchr ̈odinger operators, and it arises frequently in the study of numerous physical phenomena, from condensed matter physics to nonlinearoptics.In particular, the steady states of the Choquard equation were thoroughly investigated using a variational functional acting on the wavefunctions. In this talk I will introduce a dual formulation for the variational functional in terms of the potential induced by the wave function, and use it to explore the existence of steady states of a multi-state versionthe Choquard equation in critical and sub-critical cases.

It is known that maps in $W^{1/p,p}(S^1,S^1)$ have a well defined degree. This allows one to partition the space into disjoint classes, $W^{1/p,p}(S^1,S^1)=\bigcup_{d\in{\mathbb Z}} {\mathcal E}_d$ . It follows from a result of Brezis and Nirenberg that the $W^{1/p,p}$-distance between any two of these classes, i.e., $$ \text{dist}({\mathcal E}_{d_1},{\mathcal E}_{d_2})= \inf\{|u-v|_{W^{1/p,p}}\,:\, u\in {\mathcal E}_{d_1}, v\in {\mathcal E}_{d_2}\} $$ is zero. This reflects the fact that the degree in $W^{1/p,p}(S^1,S^1)$ is continuous, but not uniform continuous. However, in a joint work with Mironescu we proved that the distance between different classes is positive provided we assume a bound of the norm of the maps. There is also a second notion of distance which is of interest, namely $$ \text{Dist}({\mathcal E}_{d_1},{\mathcal E}_{d_2})= \sup_{u\in{\mathcal E}_{d_1}} \inf_{v\in{\mathcal E}_{d_2}} |u-v|_{W^{1/p,p}}. $$ We proved that $\text{Dist}({\mathcal E}_{d_1},{\mathcal E}_{d_2})$ equals the minimal $W^{1/p,p}$-energy in ${\mathcal E}_{d_1-d_2}$.

In this talk, I will discuss the analysis of non-simple blowup for quantized Liouville $$ -\Delta u= |x|^{2N} V(x) e^u \ \mbox{in} \ B_1$$ $$ \max u \to +\infty$$ First we show higher order vanishing of derivatives of $V$: if non-simple blowup happens then all first and second order derivatives of $V$ must vanish. For $N\geq 5$ all third order derivatives should also vanish. Second we show that all blowups for Gelfand problem $$ -\Delta u= \lambda e^u +4\pi \sum_{i} \gamma_i \delta_{p_i}$$ $$ u=0 \ \mbox{on}\ \partial \Omega$$ must be simple, even when $\gamma_i $ are integers. Finally we show that non-simple blowups are "lonely", i.e., no other blow-ups are allowed if there is one non-simple blowups. Applications to a priori estimates and degree-counting and extensions to Q-curvature problems in ${\mathbb R}^4$ will also be discussed. (Joint work with D'Aprile and L. Zhang.)

Let $n\ge2$, and let $s\in(0,1)$. Denote by $2^*_s=\dfrac{2n}{n-2s}$ the critical embedding exponent for the Sobolev--Slobodetskii space $H^s(\mathbb R^n)$. For $q\in(2,2^*_s)$, we consider the equation $$(-\Delta)^s u+u=|u|^{q-2}u\qquad\mbox{in}\ \mathbb R^n, \tag{1}$$ where $(-\Delta)^s$ is the conventional fractional Laplacian in $\mathbb R^n$ defined for any $s>0$ by the Fourier transform $(-\Delta)^s u:=F^{-1}\left(|\xi|^{2s} F u(\xi)\right)$.

Semilinear equations driven by fractional Laplacian have been studied in a number of papers. We construct several new classes of solutions to the equation (1), which, apparently, were not considered earlier.

Let $\Omega\subset\mathbb R^n$~be a convex polyhedron. For a positive sequence $R\to\! +\infty$, we define a family of expanding domains $\Omega_R=\{ x\in\mathbb R^n\,:\, x/R\in\Omega\}$ and consider the problem $$ (-\Delta)^s_{\Omega_R} u+u=|u|^{q-2}u\quad\mbox{in}\quad\Omega_R, \tag{2} $$ where $(-\Delta)^s_{\Omega_R}$ stands for some fractional Laplacian in $\Omega_R$, such as spectral fractional Dirichlet or Neumann Laplacian, etc.

Lemma 1. There exists a least energy solution u of (2), positive and smooth in $\Omega_R$.

Now we assume that the polyhedron $\Omega$ has the following property: the space $\mathbb R^n$ can be filled with its reflections, colored checkerwise. Then we can extend the function $u$ to the function $\bf u$ in the whole space by reflections consistent with the boundary conditions of $(-\Delta)^s_{\Omega_R}$.

Theorem 1. The function u is a solution of the equation (1) in $\mathbb R^n$.

In this way, we construct solutions of the equation (1) with various symmetries. Among them, there are: positive and sign-changing periodic solutions with various periodic lattices, quasi-periodic complex-valued solutions, breather-type solutions. These classes of solutions, apparently, were not studied earlier. In the local case, similar solutions are considered in [1]. However, some of our solutions are new even for $s=1$.

This talk is based on the paper [2]. A part of our results was announced in the short communication [3].

[1] L.M. Lerman, P.E. Naryshkin and A.I. Nazarov, Abundance of entire solutions to nonlinear elliptic equations by the variational method, Nonlin. Analysis, $\textbf{190}$ (2020), paper No. 111590.

[2] A.I. Nazarov and A.P. Shcheglova, Solutions with various structures for semilinear equations in } $\mathbb R^n$ driven by fractional Laplacian. Calc. Var. and PDEs, V. 62 (2023), N4, paper N112, 1--31.

[3] A.I. Nazarov and A.P. Shcheglova, New classes of solutions to semilinear equations in $\mathbb R^n$ with fractional Laplacian, $\textit{ZNS POMI}$ $\textbf{508}$ (2021), 124--133 (Russian).

Gross-Pitaevskii-Poisson (GPP) equation is a nonlocal modification of the Gross-Pitaevskii equation with an attractive Coulomb-like term. It appears in the models of self-gravitating Bose-Einstein condensates proposed in cosmology and astrophysics to describe Cold Dark Matter and Boson Stars. We investigate the existence of prescribed mass (normalised) solutions to the GPP equation, paying special attention to the shape and asymptotic behaviour of the associated mass-energy relation curves and to the limit profiles of solutions at the endpoints of these curves. In particular, we show that after appropriate rescalings, the constructed normalized solutions converge either to a ground state of the Choquard equation, or to a compactly supported radial ground state of the integral Thomas-Fermi equation. In different regimes the constructed solutions include global minima, local but not global minima and unstable mountain-pass type solutions. This is a joint work with Riccardo Molle and Giuseppe Riey.

We deal with the liquid drop model , introduced by Gamow (1928) to describe the structure of atomic nuclei. The problem consists of finding a surface $\Sigma =\partial \Omega$ in $\mathbb{R}^3$ that is critical for the following energy of regions $\Omega \subset \mathbb{R}^3$:             $$             {\mathcal E}(\Omega) = \hbox{Per }(\Omega ) + \frac 12 \int_{\Omega\times \Omega } \frac {dxdy}{|x-y|}             $$             under the volume constraint $|\Omega| = m$.             The associated Euler-Lagrange equation is             $$             H_\Sigma (x) + \int_{\Omega } \frac {dy}{|x-y|} = \lambda \quad \forall x\in \Sigma, \quad |\Omega| = m,             $$             where $\lambda$ is a constant Lagrange multiplier. Round spheres enclosing balls of volume $m$ are always solutions. They are minimizers for sufficiently small $m$. Since the two terms in the energy compete, finding non-minimizing solutions can be challenging. We find a new class of solutions with large volumes, consisting of ``pearl collars" with an axis located on a large circle, with a shape close to a Delaunay's unduloid surface with constant mean curvature. This is a joint work with Monica Musso and Andrés Zúñiga.

We consider the $n$-component Ginzburg-Landau energy \begin{equation*} \label{eq:ftnal E} E_{\varepsilon,n} (u_1,\cdots,u_n) = \frac12 \int_\Omega \sum_{j=1}^n |\nabla u_j|^2 dx + \frac{1}{4\varepsilon^2} \int_\Omega \Big(n-\sum_{j=1}^n |u_j|^2\Big)^2 dx \end{equation*} for a pair of maps $(u_1,\cdots,u_n) \in H^1_{g_1} \times\cdots\times H^1_{g_n}$ where \begin{equation*} \label{eq:deg gj} n\geq 2 \quad\hbox{and} \quad \deg (g_j,\partial\Omega)=d_j \in {\mathbb N} \text{ for } j=1,\cdots, n. \end{equation*} The purpose of this talk is to study the asymptotic behavior of minimizers $(u_{1,\varepsilon},\cdots,u_{n,\varepsilon})$ as $\varepsilon\to0$ for the functional $E_{\varepsilon,n}$. We prove that the minimizers converges locally in any $C_{loc}^k$-norm to a solution of a system of generalized harmonic map equations.