Partial Regularity for $\mathbb{A}$-quasiconvex Functionals

The analysis of functionals taking the form $\mathcal{F}=\int_{\Omega}f(\nabla u)\,\mathrm{d}x$ is a major task in the calculus of variations with a long standing tradition. Let us suppose that $\Omega\subset\mathbb{R}^{n}$ is open and bounded and that $u$ is a weakly differentiable $\mathbb{R}^{N}$-valued map. The growth condition on the integrand $f\in\mathrm{C}(\mathbb{R}^{N\times n})$ determines the functional analytic environment in which we analyse $\mathcal{F}$. A standard growth assumption, which has been studied intensively in the field, constitutes the following: There exist $p\in[1,\infty)$ and a constant $L>0$ such that for all $z\in\mathbb{R}^{N\times n}$ we have

The existence of minimisers within a given class of functions (or maps) is a fundamental question. Concretely, we want to minimise the functional $\mathcal{F}$ in $g+\mathrm{W}^{1,p}_{0}(\Omega;\mathbb{R}^{N})$ for a prescribed Dirichlet boundary datum $g\in\mathrm{W}^{1,p}(\Omega;\mathbb{R}^{N})$. In the super-linear growth case $p>1$, this task can immediately be tackled by means of the direct method, a lower semi-continuity method dating back to Tonelli. Consequently, the question arises under which assumptions on $f$ is the functional $\mathcal{F}$ sequential weakly lower semi-continuous in $\mathrm{W}^{1,p}(\Omega;\mathbb{R}^{N})$. Towards this question, convexity certainly suffices, but in the vectorial case $(N>1)$ it is seen to not be a necessary condition. Ball and Murat [8] have shown that Morrey’s notion of quasi-convexity [43], i. e., for all $z\in\mathbb{R}^{N\times n}$ and all $\zeta\in\mathrm{C}^{1}_{\mathrm{c}}(\mathbb{B};\mathbb{R}^{N})$ we have

turns out to be necessary. In fact, for unsigned integrands $f$, quasi-convexity is also a sufficient condition, as [1] have shown in [1].

Once having addressed the issue of existence of minimisers, we would like to know what further information about a minimiser we can extract. This question dates back to David Hilbert [35] and is today known by the name of regularity theory in the calculus of variations. There are many different notions of regularity and in our setting, we are interested whether a minimiser is (locally) of the class $\mathrm{C}^{1,\alpha}$. In the scalar case $(N=1)$ the notions of convexity and quasi-convexity coincide and the regularity theory, at least in the quadratic growth case, reduces to the regularity of solutions of elliptic equations established by De Giorgi [18], Nash [45] and Moser [44]. However, in the vectorial case, the regularity of minimisers can no longer be extracted from the Euler-Lagrange equation only, because, as various counterexamples show [20, 46, 29], there is no such theory for elliptic systems in general. Furthermore, full Hölder regularity can no longer be expected since minimisers may be unbounded within a small set. Adapting ideas from geometric measure theory developed by Almgren [4] and De Giorgi [19], Evans established in the non-parametric setting a fundamental partial regularity result assuming a stronger notion of quasi-convexity [27]. This means that a minimiser enjoys Hölder regularity outside a small set. We stress that partial regularity is a feature of the vectorial case. After the quadratic case, the super-quadratic case ($p\geq 2$) was established by Acerbi and Fusco in [2]. Later on, the sub-quadratic case ($1<p\leq 2$) was resolved partially by Carozza and Passarelli di Napoli in [14] and then fully by [13] in [13]. Some time later, even the Orlicz growth case has been resolved by [22] in [22]. An overview of related results can be found in [41, 42, 9, 30]. The case of linear growth for quasi-convex integrands, however, had remained an open problem, since the classical methods were bound to fail due to the lack of weak compactness. Only in the recent years, partial local Hölder regularity of the (distributional) gradient of (local) $\mathrm{BV}$-minimisers in the quasi-convex setting has been established by [33] in [33].

Let us try to roughly describe the underlying ideas on how to obtain partial Hölder regularity of the weak gradient of a minimiser $u$ of $\mathcal{F}$: The main objective is to prove a decay estimate for the excess of $u$. The excess is a quantity that, similar to Campanato’s semi-norm, measures by means of integrals the rate of oscillation of $\nabla u$. The goal is to show that if the excess is small enough, it decays with any rate $\alpha\in(0,1)$. By means of a Caccioppoli Inequality, one passes from the excess, which depends on the gradient, to a quantity depending merely on $u$. The Caccioppoli Inequality, in turn, builds on the combination of minimality and a stronger notion of quasi-convexity by means of Widman’s hole-filling trick. On the level of order $0$, the strategey then is to approximate the minimiser by a $\mathcal{B}$-harmonic map $h$, where $\mathcal{B}$ denotes a strongly Legendre-Hadamard elliptic bilinear form on $\mathbb{R}^{N\times n}$. The map $h$ has a good decay since it solves a homogeneous elliptic system. The difficulty is to construct $h$ in such a way that $u-h$ has good decay as well. Classically, the construction of $h$ follows an indirect approach utilising compactness, for example, of the embedding $\mathrm{W}^{1,2}(\mathbb{B})\to\mathrm{L}^{2}(\mathbb{B})$. This kind of approximation goes by the name of $\mathcal{B}$-Harmonic Approximation Lemma [19, 23, 25].

Many generalisations of the functional $\mathcal{F}$ have been studied. Here, we are going to replace the full gradient with a first order homogeneous differential operator $\mathbb{A}$. The two most prominent examples are the symmteric gradient $\varepsilon$ and the trace-free symmetric gradient $\tilde{\varepsilon}$. Let $V$ and $W$ be two real and finite dimensional Hilbert spaces. We are going to consider differential operators of the form

For $\xi\in\mathbb{K}^{n}$, $\mathbb{K}=\mathbb{R},\mathbb{C}$, the linear map $\mathbb{A}[\xi]v=\sum_{\alpha=1}^{n}\xi_{\alpha}\mathbb{A}_{\alpha}v$, modulo a factor of $-i$, is called the symbol map associated to the differential operator $\mathbb{A}$. We say that $\mathbb{A}$ is $\mathbb{K}$-elliptic if the symbol map $\mathbb{A}[\xi]$ is one-to-one for all $\xi\in\mathbb{K}^{n}\setminus\{0\}$ ([50, 49, 36]). The notion of $\mathbb{R}$-ellipticity has been characterised by means of Fourier multipliers [40] and singular integrals [11] that the Korn-type Inequality

is satisfied by the differential operator $\mathbb{A}$. In the super-linear growth case, Conti and Gmeineder showed that this allows to reduce the question of partial Hölder regularity of a local minimiser of a functional of the form

where $f\in\mathrm{C}(W)$ is of $p$-growth ($p>1$), to the full gradient case [17]. Ornstein’s Non-Inequality [47, Theorem 1], [16, 37], stating that there is no non-trivial Korn Inequality in the $\mathrm{L}^{1}$-setting, implies that such a reduction is impossible in the linear growth regime. Consequently, it had constituted a highly non-trivial task to adapt the full gradient case [33] to the symmetric gradient case [32] in the linear growth regime.

In the convex case, partial regularity for linear growth functionals has been known in the convex context following the work of [6] [6] (also see [48, 31] for variations of this theme). However, the methods employed therein are confined to the convex case. In the linear growth context, the key difficulty to overcome is the lack of weak compactness. This concerns the existence of minimisers as much as their regularity theory. In particular, this excludes indirect methods like the by now classical $\mathcal{B}$-Harmonic Approximation Lemma [23, 25, 24]. A direct approach was needed to construct a $\mathcal{B}$-harmonic approximation. [33] solved this problem by showing that the traces of $\mathrm{BV}$-maps on spheres of radius $R$ enjoy for $\mathcal{L}^{1}$-almost all sufficiently small radii $R$ more regularity than the default $\mathrm{L}^{1}$-regularity. This is called a Fubini-type property of $\mathrm{BV}$-maps and it was in effect the key point to construct a $\mathcal{B}$-harmonic approximation by solving the elliptic system

We note that the solution operator

associated to this system cannot be bounded as an operator from $\mathrm{L}^{1}$ to $\mathrm{W}^{1,1}$. In other words, without more regularity of $\operatorname{Tr}_{B_{R}(x_{0})}(u)$ we lack tools to precisely measure how close the $\mathcal{B}$-harmonic map $h$ is to $u$. This is why the Fubini-type property of $\mathrm{BV}$-maps is essential for a direct approach in the linear growth regime.

[32] was able to adapt the ideas used in [33] to the scenario where the full gradient is replaced by the symmetric gradient [32]. The main difficulties were to prove a Fubini-type property and, building on the latter, to prove precise estimates for $u-h$, where $h$ denotes a suitable $\mathcal{B}$-harmonic approximation of $u$.

Our scope is to show that the method for the symmetric gradient case extends to an entire class of first order homogeneous differential operators with constant coefficients, namely the class of $\mathbb{C}$-elliptic operators.

In line with the full [33] and symmetric gradient case [32], we will work from now on under the following assumptions:

• (H0)

  The differential operator $\mathbb{A}$ is $\mathbb{C}$-elliptic.

• (H1)

  The integrand $f\in\mathrm{C}^{2,1}_{\mathrm{loc}}(W)$ is of linear growth.

• (H2)

  The integrand $f$ is strongly $V_{1}$-$\mathbb{A}$-quasi-convex, where $V_{1}$ denotes the reference integrand to be defined in the the upcoming section on preliminaries: There exists $\nu>0$ such that $F=f-\nu V_{1}\circ|\cdot|$ is $\mathbb{A}$-quasi-convex, i. e., for all $w\in W$ and all $\zeta\in\mathrm{C}^{\infty}_{\mathrm{c}}(\mathbb{B})$ we have

  $$F(w)\leq\fint_{\mathbb{B}}F\left(w+\mathbb{A}\zeta\right)\,\mathrm{d}x.$$

For $\omega\subset\mathbb{R}^{n}$ open and bounded we associate to the integrand $f$ the functional $\mathcal{F}[u;\omega]=\int_{\omega}F(\mathbb{A}u(x))\,\mathrm{d}x$, where $u\in\mathrm{W}^{\mathbb{A},1}(\omega)$.

We recall that to any $\mathbb{R}$-elliptic potential $\mathbb{A}$ exists an annihilator, see [51], $\mathcal{A}=\sum_{|\alpha|=k}\mathcal{A}_{\alpha}\partial_{\alpha}$ with $\mathcal{A}_{\alpha}\in\mathcal{L}(W;V)$ such that the symbol complex

is exact for all $\xi\in\mathbb{R}^{n}\setminus\{0\}$. This shows that the notion of $\mathbb{A}$-quasi-convexity is equivalent to the more widely known notion of $\mathcal{A}$-quasi-convexity which is strongly linked to weak sequential lower semi-continuity of the functional $\mathcal{F}$, which Fonseca and Müller showed in [28].

We fix $\Omega\subset\mathbb{R}^{n}$ an open and bounded Lipschitz domain and $g\in\mathrm{W}^{\mathbb{A},1}(\Omega)$ a prescribed boundary datum. Due to the lack of weak compactness, analogously to the full gradient case, the task to minimise $\mathcal{F}$ within a given Dirichlet class $g+\mathrm{W}^{\mathbb{A},1}_{0}(\Omega)$ cannot be tackled by plainly applying the direct method. Hence, we pass from the Sobolev-type space $\mathrm{W}^{\mathbb{A},1}$ to the $\mathrm{BV}$-type space $\mathrm{BV}^{\mathbb{A}}$, hoping for better compactness with respect to the weak^∗-topology on the latter space. Already in the full gradient case, this forces us to somehow relax our functional $\mathcal{F}$ to this larger space $\mathrm{BV}^{\mathbb{A}}$. Without the assumption of $\mathbb{C}$-ellipticity, this relaxation procedure cannot be implemented analogously: We recall that in the full gradient case, Alberti’s rank-one result [3] for $\mathrm{BV}$-maps has proven to be essential in order to obtain an integral representation of the Lebesgue-Serrin extension [5]. Using that quasi-convexity implies rank-one convexity, Alberti’s result ensures that the strong recession function of a quasi-convex integrand with linear growth is well-defined on the rank-one cone. The requisite result paralleling Alberti’s result in the $\mathbb{R}$-elliptic case has been established by [21] in [21]. Weak^∗-compactness of closed, norm bounded sets in $\mathrm{BV}^{\mathbb{A}}$ as well as the existence of a strictly continuous and linear trace operator $\operatorname{Tr}_{\Omega}\colon\mathrm{BV}^{\mathbb{A}}(\Omega)\to\mathrm{L}^{1}(\partial\Omega)$ for open and bounded Lipschitz domains are two features exclusive to the $\mathbb{C}$-elliptic case [10, Theorem 1.1], [34, Theorem 1.1]. Since the trace-operator on $\mathrm{BV}^{\mathbb{A}}$ is discontinuous with respect to weak^∗-convergence, the boundary condition is no longer reflected by the space $\mathrm{BV}^{\mathbb{A}}$ but rather by the relaxed functional itself. The boundary condition then is encoded by a so called penalty term, which already pops up in the full gradient case [38]:

where $f^{\infty}(w)=\limsup_{t\to\infty}\frac{f(tw)}{t}$ denotes the strong recession function and $\xi\otimes_{\mathbb{A}}v=\sum_{\alpha=1}^{n}\xi_{\alpha}\mathbb{A}_{\alpha}v$ for $\xi\in\mathbb{R}^{n}$ and $v\in V$. We stress that it is necessary to assume that $\mathbb{A}$ is $\mathbb{C}$-elliptic in order for this integral expression to be well-defined. The relaxed functional then takes the form ([10, Section 5], [7])

We note that (H2) implies that there exist $b\in\mathbb{R}$ and $c>0$ such that for all $\zeta\in g+\mathrm{W}^{\mathbb{A},1}_{0}(\Omega)$ we have $\mathcal{F}[\zeta;\Omega]\geq cV_{1}(\fint_{\Omega}|\mathbb{A}\zeta|\,\mathrm{d}x)+b$. This can be inferred from an extension and gluing argument similar to the argument in [15, 218–219]. Identifying $\overline{\mathcal{F}}_{g}[u;\Omega]$ as the Lebesgue-Serrin extension [10, Section 5]

yields coercivity of the relaxation. Hence, under our assumption, generalised minimisers subject to a given Dirichlet boundary condition exist by means of the direct method. Since we are only striving for a local regularity result, it is natural to consider the class of local generalised minimisers:

At this stage, we are ready to formulate the main theorem:

We wish to point out that for the present paper, the assumption of $\mathbb{C}$-ellipticity is crucial and visible on several stages (so e. g. in the very definition of the functionals where boundary traces come into play); the elliptic case, however, seems to require refined methods.

For a finite dimensional real vector space $Z$ we use the shorthand notation $d_{Z}=\dim_{\mathbb{R}}Z$. Furthermore, we will suppress the target vector space when dealing with different function spaces. For example, if $u\colon\Omega\to Z$ is a $\mathrm{L}^{1}$-map, we simply write $u\in\mathrm{L}^{1}(\Omega)$ instead of $u\in\mathrm{L}^{1}(\Omega;Z)$. Within the context, it will be clear which target vector space we are referring to. Furthermore, by $|\cdot|$ we will denote any norm of a finite dimensional, normed vector space such as $\mathcal{L}(V;W)$, $V$ or $\mathcal{L}(V\times V;W)$. Since all norms of a finite dimensional normed vector space are equivalent, this is an non-problematic convention. Throughout, we fix an orthonormal basis $(v_{j},...,v_{N})$ of $V$, i. e., $N=d_{V}$. Furthermore,

denotes the standard basis of $\mathbb{R}^{N\times n}$.

Integration with respect to the $(n-1)$-dimensional Hausdorff-measure $\mathcal{H}^{n-1}$ will be denoted by $\,\mathrm{d}\sigma_{x}$, where $x$ is integration variable.

As usual, $B_{r}(x_{0})\subset\mathbb{R}^{n}$ denotes the open ball with centre $x_{0}$ and radius $r>0$. Often we will suppress the centre of the ball if it is clear within the context and we will simply write $B_{r}$. Furthermore, we put $\mathbb{B}=B_{1}(0)$ and $\mathbb{S}=\partial\mathbb{B}$.

By $\mathcal{M}(\Omega;Z)$ we denote the space of $Z$-valued finite Radon measures on $\Omega$. For $\mu\in\mathcal{M}(\Omega)$ and open, bounded subsets $\omega\subset\Omega$, the total variation-measure of $\mu$ will be denoted by $|\mu|$ and the average of $\mu$ on $\omega$ with respect to the Lebesgue measure will be written as $(\mu)_{\omega}\coloneq\frac{\mu(\omega)}{\mathcal{L}^{n}(\omega)}.$

Let $V_{1}(t)=\sqrt{1+t^{2}}-1$ denote the reference integrand. We will use the shorthand notation $V_{1}(z)=V_{1}(|z|)$.

For a $\mathrm{C}^{1}$-function $G\colon Z\to\mathbb{R}$ and $z_{0}\in Z$ we denote by

the linearisation of $G$ at $z_{0}$.

For $k\in\mathbb{N}$ we denote by $\mathcal{P}_{k}(Z)$ the vector space of all polynomials

of degree at most $k$.

By $C$ we will denote a generic constant which may vary from line to line. Since it is very important throughout on which parameters a constant depends on, we will write for example $C(M,p)$ if the constant depends on $M$ and $p$.

For $\xi\in\mathbb{R}^{n}$ and $v\in V$ we put $\xi\otimes_{\mathbb{A}}v=\sum_{\alpha=1}^{n}\xi_{\alpha}\mathbb{A}_{\alpha}v$. Furthermore, we denote by

the effective range of $\mathbb{A}$ and we call

the null-space of $\mathbb{A}$. The formally adjoint operator $\mathbb{A}^{*}$ is here defined by the formula

Let $\Omega\subset\mathbb{R}^{n}$ be open and let $\mathbb{M}\subset\mathbb{R}^{n}$ be an embedded $(n-1)$-dimensional $\mathrm{C}^{1}$-submanifold of $\mathbb{R}^{n}$. For $\alpha\in(0,1)$ and $p\in[1,\infty)$, we recall the definition of the fractional Sobolev space (semi)-norms on $\Omega$ and $\mathbb{M}$, respectively:

• •

  $[u]_{\mathrm{W}^{\alpha,p}(\Omega)}=\Big{(}\int_{\Omega^{2}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+\alpha p}}\,\mathrm{d}x\,\mathrm{d}y\Big{)}^{\frac{1}{p}}$, $\left\lVert u\right\rVert_{\mathrm{W}^{\alpha,p}(\Omega)}=\left\lVert u\right\rVert_{\mathrm{L}^{p}(\Omega)}+[u]_{\mathrm{W}^{\alpha,p}(\Omega)}$,

• •

  $[u]_{\mathrm{W}^{\alpha,p}(\mathbb{M})}=\Big{(}\int_{\mathbb{M}^{2}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n-1+\alpha p}}\,\mathrm{d}\sigma_{x}\,\mathrm{d}\sigma_{y}\Big{)}^{\frac{1}{p}}$, $\left\lVert u\right\rVert_{\mathrm{W}^{\alpha,p}(\mathbb{M})}=\left\lVert u\right\rVert_{\mathrm{L}^{p}(\mathbb{M})}+[u]_{\mathrm{W}^{\alpha,p}(\mathbb{M})}$.

We are going to collect prerequisites on $\mathbb{A}$-weakly differentiable maps. In the spirit of [10], we define Sobolev- and $\mathrm{BV}$-type spaces as follows:

Let $u\in\mathrm{BV}^{\mathbb{A}}_{\mathrm{loc}}(\Omega)$. Then we consider the Radon-Nikodým decomposition of $\mathbb{A}u$ with respect to the Lebesgue measure $\mathbb{A}u=\mathbb{A}^{a}u+\mathbb{A}^{s}u$, where $\mathbb{A}^{a}u$ denotes the absolutely continuous part and $\mathbb{A}^{s}u$ the singular part. Next, we recall different notions of convergence in $\mathrm{BV}^{\mathbb{A}}$:

The proof of the last assertion is analogous to the $\mathrm{BD}$-case [32].

We will use the shorthand notation $\fint_{\Omega}g(\mathbb{A}u)=\frac{\int_{\Omega}g(\mathbb{A}u)}{\mathcal{L}^{n}(\Omega)}$.

The proof is analogous to the proof of [33, Lemma 4.2] and [32, Lemma 5.1].

The Caccioppoli Inequality is indispensable for our proof of partial regularity. However, it is line for line analogous to the full and symmetric the gradient case [33, Proposition 4.3], [32, Proposition 5.2], replacing $\nabla$ or $\varepsilon$ with $\mathbb{A}$, respectively: by exploiting the strong $V_{1}$-$\mathbb{A}$-quasi-convexity and minimality of $u$, we apply Widman’s hole-filling trick and iterate the resulting inequality.