Quasiconvex bulk and surface energies: C1,α regularity
-
Menita Carozza
,
Luca Esposito
Abstract
We establish regularity results for equilibrium configurations of vectorial multidimensional variational problems, involving bulk and surface energies. The bulk energy densities are uniformly strictly quasiconvex functions with
1 Introduction and statements
In this article, we study multidimensional vectorial variational problems involving bulk and surface energies, mainly related to problems issuing from material science and computer vision. Namely, we deal with regularity properties of solutions to such problems. The model problem
where
with the constraints
In the cited articles, it is proved
where
then
More recently, De Philippis and Figalli [17] and Fusco and Julin [27], improved this result showing that minimal configurations of Dirichlet-type functional (1.1) satisfy
for some
The case of general functionals with
Differently from the scalar case, in the vectorial setting, only a few regularity results for minimizers of integral functionals involving both bulk and interfacial energies are available in the literature. According to our knowledge, the only articles dealing with the vectorial case are [6] and [11]. In [6], the regularity for vector-valued free interface variational problems is treated within the context of
where
for every
Under these assumptions, the authors proved the existence of local minimizers for the functional (1.3), for any
In this article, we generalize the results given in [11] under two viewpoints. First, we treat the more general case of
In the rest of this article, we focus our attention on integral functionals defined as follows:
where
We assume that
for every Borel set
with
In the early stages, the studies in this area had been carried out in the setting of varifolds and currents. These results can be applied to surfaces of arbitrary codimension, but with rather strong regularity assumptions on the integrands of the anisotropic energies [9,39].
More recently, the regularity assumptions on the integrands
In the vectorial setting debated in this article, where the bulk energy is of general type with
We say that a pair
Existence and regularity of local minimizers of integral functionals of the type
with uniformly strict
In order to prove the existence of local minimizers for functionals involving both bulk and surface energies of general type (1.4), we invoke a result stated in [11]. The only difference in our setting is the presence of the anisotropic term
Theorem 1.1
Let
In particular,
Then, we obtain
Theorem 1.2
Let
then there exists an open set
The proof of the previous result is based on a comparison argument with solutions of a suitable linearized system. We establish decay estimates for the “hybrid” excess functions
2 Notation and preliminary results
Let
We denote by
We omit the dependence on the center when it is clear from the context:
for the usual inner product of
If
for
It is well known that for quasiconvex
for all
Furthermore, if
for all
Proposition 2.1
Let
for every
for all
where
The next iteration lemma has important applications in the regularity theory (for its proof, we refer to [32, Lemma 6.1]).
Lemma 2.2
Let
where
Given a
We state the following lemma about the growth of
Lemma 2.3
Let
for any
for all
2.1 Sets of finite perimeter
If
Given a Lebesgue measurable set
Moreover, we denote the perimeter of
The support of
(see [37, Proposition 12.19]). It holds that
exists and belongs to
Federer’s criterion, see for instance [37, Theorem 16.2], ensures that
Remark 2.4
(Minimal topological boundary) If
Therefore, by (2.3),
Finally, by De Giorgi’s rectifiability theorem (see [37, Theorem 15.9]), we obtain
on Borel sets compactly contained in
It is well known that if
Proposition 2.5
(Gauss-Green measure and set operations) If E and F are the sets of finite perimeter, it shows that
then
2.2 Anisotropic surface energy
Definition 2.6
(Elliptic integrands) Given an open set
for every Borel set
Remark 2.7
(Comparability to perimeter) In order to prove the regularity of minimizers of anisotropic surface energies, it is well known that (see the seminal article [3]) a
for any
for any set
Proposition 2.8
Let
Proof
Let us observe that since
Using (2.6), we deduce
3 Lower semicontinuity
In this section, we prove Theorem 1.1. We base ourselves on the proof given in [11], focusing our attention on the only difference we have, the presence of the anisotropic surface energy. We obtain a semicontinuity result for the anisotropic perimeter, which is essential to handle this novelty.
Given a one-homogeneous Borel function
where
Theorem 3.1
(Reshetnyak’s lower semicontinuity theorem) Let
for every lower semicontinuous function
We are now able to prove the following result.
Proposition 3.2
Let
whenever
Proof
We recall that if
Thus, the claim follows by virtue of Reshetnyak’s lower semicontinuity theorem.□
Proof of Theorem 1.1
We omit the proof as it can be obtained following verbatim the arguments used in [11, Section 3]. The only difference with the proof given in [11] concerns the presence of the anisotropic perimeter. In this regard, we can use the lower semicontinuity result given earlier. We mention that the convergence
4 Higher integrability result
This section is devoted to the proof of a higher integrability result for the gradient of the function
Theorem 4.1
Assume that (
where
Proof
We consider
Setting
by the uniformly strict quasiconvexity of
We write
where we used the fact that
where we used assumptions (
and, by Lemma 2.2, we deduce that
The Sobolev-Poincaré inequality (see [32, p. 102]) implies that
and the conclusion follows by virtue of Giaquinta-Modica theorem (see [32, p. 203]).□
5 Decay estimates
In this section, we collect some energy estimates for minimizers of the functional (1.4) that will be crucial in the proof of Theorem 1.2. In order to obtain them, we will employ a well-known blow-up technique involving a quantity called excess, which includes all the energy terms of the functional. We have to use different types of excess depending on whether the assumption
for
Proposition 5.1
Let
then
Proof
In order to prove (5.3), we argue by contradiction. Let
and
The constant
Step 1. Blow-up.
Set
One can easily check that
Set
Note that
It follows that
Therefore, by (5.4) and (5.8), there exist a (not relabeled) subsequence of
where we used the fact that
Therefore, by the relative isoperimetric inequality,
In the sequel, the proof will proceed differently depending on whether
The first case is easier to handle. To understand the reason, let us introduce the expansion of
for any
We claim that
for any
and (5.14) follows by the definition of
In the second case, the suitable rescaled functional to consider in the blow-up procedure is
Then, we claim that
for all
where we used that the last integral vanishes outside the support of
or, equivalently, using the definitions of
where
by the definition of
i.e., (5.15).
Step 2. A Caccioppoli-type inequality.
We claim that there exists a constant
We divide the proof into two substeps.
Substep 2.a
The case
Consider
for any
for any
and so the constant in (2.2) can be taken independently of
Set
By the uniformly strict quasiconvexity of
We estimate separately the two addends in the right-hand side of the previous chain of inequalities. We deal with the first addend by means of a rescaling of the minimality condition of
i.e., by the definitions of
for
Choosing
where we used Lemma 2.3 and the second estimate in (2.1), and the fact that
Hence, combining (5.21) with (5.22) and (5.23), using the properties of
where we used Young’s inequality. By adding
Therefore, by the definition of
which, by the relative isoperimetric inequality and the hypothesis of this substep, i.e.,
Substep 2.b
The case
As in the previous substep, we fix
for any
We remark that Lemma 2.3 applies to
and, by the uniformly strict quasiconvexity conditions (
for all
Set
By (5.25) and since
As in the previous step, we estimate separately the two addends in the right-hand side of the previous chain of inequalities. We deal with the first addend by means of a rescaling of the minimality condition of
or, equivalently, by the definition of
Choosing
By remarking that
for every
Now, we estimate the second addend in the right-hand side of (5.26). Using the upper bound on
Inserting (5.28) and (5.29) into (5.26), we infer that
Using the hole-filling technique as in the previous case, we obtain
The assumption
So, by virtue of Lemma 2.2, from the previous estimate, we deduce that
Therefore, by the definition of
which, by the relative isoperimetric inequality and since we have
Step 3.
We prove that there exists a constant
for any
It will follow that
since
As mentioned earlier, we will divide the proof in two substeps.
Substep 3.a
The case
We claim that
for all
for every
We divide by
We partition the unit ball as
By (5.8), we obtain
We rewrite (5.32) as follows:
By virtue of the first estimate in (2.1) and Hölder’s inequality, we obtain
thanks to (5.4) (to bound
By (5.4) and the definition of
where we used (5.8). Since, by (5.9),
Note that (5.33) yields that
Thus, by the weak convergence of
By the second estimate in (2.1), we deduce that
where we used (5.4) in the second inequality to bound
and so,
By (5.35), (5.37), (5.38), and (5.39), passing to the limit as
and with
i.e.,
since
Substep 3.b
The case
We claim that
for all
for every
Dividing the previous inequality by
By the definition of
Just as earlier, we partition
and we write
Arguing as in (5.35) , we obtain that
Moreover, we have that
where we used (5.8). Since
and we obtain
By (5.41)–(5.43), passing to the limit as
and with
asserting the claim. By Proposition 2.1, we deduce also in this case that
Step 4. An estimate for the perimeters
Our aim is to show that there exists a constant
By the minimality of
Using the change of variable
where
for every Borel set
By coarea formula, the relative isoperimetric inequality, the choice of the representative
Therefore, we may choose
We remark that Proposition 2.8 holds also for
By the choice of the representative of
On the other hand, by (2.8) and the additivity of the measure
since
where we used (5.8). The previous estimate leads to (5.44). We reach the same conclusion if
Step 5. Conclusion.
By the change of variable
where we used (5.8) and estimate (5.50). We remark that
being
There exists
Thus, we interpolate by Hölder’s inequality, and we obtain
On the one hand, by the Poincaré-Wirtinger inequality and (5.31), we obtain
On the other hand, by the Sobolev-Poincaré inequality, we infer
where we used (5.30) and (5.8). Therefore, (5.51) follows at once.
By virtue of the strong convergence of
The contradiction follows, by choosing
Next, we prove a suitable decay estimate that allows us to prove Theorem 1.2 without the assumption
where
In the proof of Proposition 5.2, we will only elaborate the steps substantially different from the corresponding ones in the proof of Proposition 5.1.
Proposition 5.2
Let
then
Proof
In order to prove (5.53), we argue by contradiction. Let
and
The constant
Step 1. Blow-up.
We set
One can easily check that
Let us note that
It follows that
Therefore, by virtue of (5.54), (5.56), and (5.57), there exist a (not relabeled) subsequence
where we used the fact that
since
Therefore, by the relative isoperimetric inequality,
Step 2. A Caccioppoli-type inequality.
We claim that there exists a constant
We divide the proof into two substeps.
Substep 2.a
The case
The proof of this substep goes exactly as that of Substep 2.a of Proposition 5.1 up to estimate (5.24). Next, we observe that
and this, by the relative isoperimetric inequality, yields the estimate (5.62).
Substep 2.b
The case
We fix
for any
In order to estimate the first addend of the right-hand side of the previous inequality, we recall that
We remark that the reverse Hölder inequality stated in Theorem 4.1, through the change of variable
By Hölder’s inequality and inserting the previous inequality into estimate (5.64), we obtain
where we used the fact that
Thanks to the Lemma 2.3, Young’s inequality, and the properties of
Using the hole-filling technique as in the proof of the previous theorem, we obtain
By virtue of the iteration Lemma 2.2, the previous estimate gives
where
which, by the relative isoperimetric inequality and since
The proofs of Step 3 and Step 4 of Proposition 5.1 hold true also in this case.
Step 5. Conclusion
The change of variable
where
where
6 Proof of the main theorem
Here, we give the proof of Theorem 1.2 through a suitable iteration procedure. It is easy to obtain the following Lemmata, arguing exactly in the same way as in [11, Lemma 6.1].
Lemma 6.1
Let
where
Lemma 6.2
Let
where
6.1 Proof of Theorem 1.2
Proof
We consider the set
and let
for every
where we used Jensen’s inequality to estimate
The previous estimate implies that
where
The last inequality implies, by the Campanato characterization of Hölder continuous functions (see [32, Theorem 2.9]), that
When the assumption
Acknowledgements
The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). M. Carozza acknowledged support by GNAMPA.
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Author contributions: All authors have accepted their responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. The authors MC, LE and LL contributed equally to the entire manuscript.
-
Conflict of interest: The authors state no conflict of interest.
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Data availability statement: Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.
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- Existence and uniqueness of solution for a singular elliptic differential equation
- The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
- Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
- Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions
- Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
- Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions
- Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator
- On a nonlinear Robin problem with an absorption term on the boundary and L1 data
- Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities
- Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
- Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3
- Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation
- Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
- A minimization problem with free boundary for p-Laplacian weakly coupled system
- Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
- k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
- Infinitely many solutions for Hamiltonian system with critical growth
- On optimal control in a nonlinear interface problem described by hemivariational inequalities
- Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
- Potential and monotone homeomorphisms in Banach spaces
- Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
- Supercritical Hénon-type equation with a forcing term
- Concentration of blow-up solutions for the Gross-Pitaveskii equation
- Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
- Regularity for critical fractional Choquard equation with singular potential and its applications
- Double phase anisotropic variational problems involving critical growth
- Normalized solutions of NLS equations with mixed nonlocal nonlinearities
- Nontrivial solutions for resonance quasilinear elliptic systems
- Standing waves for Choquard equation with noncritical rotation
- Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
- Modified quasilinear equations with strongly singular and critical exponential nonlinearity
- Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
- Nests of limit cycles in quadratic systems
- Regularity of minimizers for double phase functionals of borderline case with variable exponents
- Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
- Ground state solutions for magnetic Schrödinger equations with polynomial growth
- Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
- Dynamics for wave equations connected in parallel with nonlinear localized damping
- Boussinesq's equation for water waves: Asymptotics in Sector I
- Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
- Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
- Normalized solutions for the double-phase problem with nonlocal reaction
- Normalized solutions for Sobolev critical fractional Schrödinger equation
- Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
- Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
- Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
- Quasiconvex bulk and surface energies: C1,α regularity
- Time decay estimates of solutions to a two-phase flow model in the whole space
- Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
- The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
- A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
- Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
- Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
- The second Yamabe invariant with boundary
- Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
- Normalized solutions for the Choquard equations with critical nonlinearities
- Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
- Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
- Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
- Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
- Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
- Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
- Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
- On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
- Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
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Uniqueness and nondegeneracy of ground states for
- Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
- Weighted Hardy-Adams inequality on unit ball of any even dimension
- Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
- Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
- Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
- The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
- Nonlocal heat equations with generalized fractional Laplacian
- Choquard equations with recurrent potentials
- Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
- The Riemann problem for two-layer shallow water equations with bottom topography
- Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
- On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
- Energy-variational solutions for viscoelastic fluid models
- Stability on 3D Boussinesq system with mixed partial dissipation
- Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
- Erratum
- Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
