Regularity results for an optimal design problem with lower order terms
-
Luca Esposito
and
Lorenzo Lamberti
Abstract
We study the regularity of the interface for optimal energy configurations of functionals involving bulk energies with an additional perimeter penalization of the interface. Here we allow a more general structure for the energy functional in the bulk term. For a minimal configuration
1 Introduction and statements
In this paper we will deal with energy functionals of the type
where
where
In 1993 L. Ambrosio and G. Buttazzo in [2] proved that if
where
In the paper by F. H. Lin (see [20]) it was proved that
for some
In 1999 F.H. Lin and R.V. Kohn in [21] treated a more general quadratic bulk energy of the type (1.1) actually proving the same regularity proved by the first author for the model case (1.2). Indeed, they proved that the singular part
Throughout the paper we will assume that the density energies F and G in (1.1) satisfy the following assumptions:
Concerning the coefficients we assume that
We will denote by
and the same holds true for
Moreover, to ensure the existence of minimizers we assume the boundedness of the coefficients and the ellipticity of the matrices
where ν, N and L are three positive constants. We are interested in the regularity of minimizers of the following constrained problem.
Definition 1.
We shall denote by ((${\mathrm{P}_{\mathrm{c}}}$)) the constrained problem
where
The problem of handling with the constraint
for some suitable
Definition 2.
We shall denote by (($\mathrm{P}$)) the penalized problem
where
From the point of view of regularity, the extra term
Theorem 1.
Let
there exists a relatively open set
there exists
For the reader’s convenience the paper is structured in sections which reflect the proof strategy. Section 2 collects known results and preliminary definitions. As in the case of minimizers of the Mumford–Shah functional, the proof of regularity is based on the study of the interplay between the perimeter and the bulk energy (see [3, 20]).
We point out that the Hölder exponent
either in the case that
or in the case that there exists an half-space H such that
for some
are Hölder continuous,
decays as
Indeed, we prove the height bound lemma and the Lipschitz approximation theorem. In this section we also compute the Euler–Lagrange equation for
2 Preliminary notations and definitions
In the rest of the paper we will write
For any
The following definition is standard.
Definition 3.
Let
Furthermore, we say that v is a local minimizer of the integral functional
for all
It is clear that any minimizer u of problem ((${\mathrm{P}_{\mathrm{c}}}$)) is a local minimizer of the functional (1.1) and therefore satisfies the Euler–Lagrange equation
It is worth mentioning that for a quadratic integrand
Lemma 1.
Let
with
for some
Proof.
The proof of this lemma is standard and can be found in [14]. Inspecting the proof in [14] one can also obtain an explicit expression of the constant
The next lemma can be found in [3, Lemma 7.54].
Lemma 2.
Let
for some constants
2.1 From constrained to penalized problem
The next theorem allows us to overcome the difficulty of handling with the constraint
Theorem 2.
There exists
for some
Proof.
The proof can be carried out as in [10, Theorem 1]. For the reader’s convenience we give here its sketch, emphasizing main ideas and minor differences with respect to the case treated in [10].
The first part of the theorem can be proved by contradiction. Assume that there exist a sequence
Without loss of generality we may assume that
By condition (2.4), it follows that the sequence
Step 1: Construction of
where
Let us assume, without loss of generality, that
Let us now define the following bi-Lipschitz function used in [10] which maps
for some
we have
(2.7)
Step 2: Estimate of
It is clear from this expression that, since
It is not difficult to find out also that
Concerning
Using the fact that
provided that we choose
Summarizing we gain the following inequalities for the Jacobian of Φ:
Now we can perform the change of variables
The two terms
for any
We proceed estimating the first difference in the previous equality, being the other similar and indeed easier to handle:
The first term
If we apply the last inequality to the vectors
we are led to estimate
Therefore we have
In addition, choosing
Summarizing we finally get
for some constant
where Θ is defined in (2.4). The second term
In conclusion, since the other terms in (2.12) can be estimated in the same way, collecting estimates (2.13) and (2.14), we get
Since the same estimate holds true for
for some constant
Step 3: Estimate of
Thereafter we can estimate
We address the reader to [10] where explicit calculations are given. In order to estimate
Notice that the last integral in the above formula is non-negative since Φ is a contraction in
thus concluding that
Step 4: Estimate of
Therefore, if we choose
for some positive
if
The previous theorem motivates the following definition.
Definition 4 (Λ-minimizers).
The energy pair
whenever
3 Decay of the bulk energy
In the first part of this section we collect some preliminary results concerning decay estimates for local minimizers u of the functional (1.1) when E is fixed. We start quoting higher integrability results both for local minimizers of functional (1.1) and for comparison functions that we will use later in the paper. It is worth mentioning that the following lemmata can be applied in general to minimizers of integral functionals of the type
assuming that the energy density only satisfies the structure condition (1.3) and the growth conditions (1.6) and (1.7), without assuming any continuity on the coefficients. Therefore, functionals of the type (1.1) belong to this class and in addition the involved estimates only depend on the constants appearing in (1.6) and (1.7) but do not depend on E accordingly.
Lemma 3.
Let
where
Proof.
Without loss of generality we may assume that
Testing the minimality of u with v and using growth condition (2.2) we deduce that
Adding to both sides
Thus we get
where
with
Iterating the previous estimate using Lemma 1, we deduce that there exists a constant
We get the thesis if we divide by
Remark 1.
We observe that the reverse Hölder inequality stated in the previous lemma can be also proved exactly in the same way replacing the balls with the cubes. The reverse Hölder inequality written on cubes is the suitable version in order to apply Calderòn–Zygmund decomposition and Gehring’s lemma (see [14, Proposition 6.1]), thus obtaining the higher integrability estimate on cubes. Finally, the higher integrability estimate on balls stated below, which is suitable in our setting, can be deduced by a covering argument (see [13, Appendix]).
Lemma 4.
Let
where
In the next subsection we will prove some energy density estimates by using a standard comparison argument. For this purpose we will need a reverse Hölder inequality for the comparison function defined below.
Definition 5 (Comparison function).
Let
where
For the comparison function v defined in (3.3) we can state the following reverse Hölder inequality up to the boundary of
Lemma 5.
Let
Let
where
Proof.
Let
In case (i) we can proceed exactly as in Lemma 3 to get the desired estimate. Let us then consider case (ii) which is slightly different. Choose
and
We start comparing the energy of φ and v inside a generic set
Using the growth conditions (1.6) and (1.7) and Young’s inequality, we deduce that
Recalling the growth condition (2.2), we estimate
We can conclude from (3.5) that
We compute now the previous integrals on the set
Finally, we can resume (3.6) to conclude this first part concerning the energy estimate of φ, using again (2.2),
We can summarize the previous estimate as follows:
Now we observe that
(3.8)
Now we use the “hole-filling” technique adding
where
Therefore, being
for some universal constant
Finally, we can apply reverse Hölder inequality (3.2) for u in the last estimate to get (3.4). ∎
Reasoning in a similar way as above, the higher integrability for v can be obtained by means of Gehring’s lemma (see [14, Proposition 6.1]).
Lemma 6.
Let
where
3.1 A decay estimate for elastic minima
In this subsection we prove a decay estimate for elastic minima that will be crucial for the proof strategy.
Indeed, we show that if
for any
Theorem 3.
Let
As a consequence of Theorem 3, using Poincaré’s inequality and the characterization of Campanato spaces (see for example [14, Theorem 2.9]), we can infer that
Remark 2.
Let
Notation 4.
In the sequel
If
We will denote
In order to prove the main lemma of this section we introduce the following preliminary result. For the reader’s convenience we give here a sketch of the proof, which can be found in [23]. Actually we state here a weaker version that is suitable for our aim.
Lemma 7.
Let
where
for some
and
for some
Then
Proof.
Fix
Notice that by assumption
Let w be the solution of
The last equation can be rewritten as
where
where
Step 1: Tangential derivatives of w. Let us denote with τ the general direction tangent to the hyperplane
Hence, Caccioppoli’s inequality holds:
for all balls
for any
Step 2: Regularity of
which implies again by Step 1
We can conclude that
for any
for any
(3.15)
Step 3: Comparison between v and w. Subtracting the equation for w from the equation for v we get
for any
Finally, we can estimate
for any
by (3.14), (3.15), the minimality of w and Young’s inequality we gain
which leads to our aim if we apply Lemma 2. ∎
The next lemma is inspired by [13, Proposition 2.4] and is the main result of this section.
Lemma 8.
Let
there exists a halfspace H such that
then
for some positive constant
Proof.
Let us fix
and
Let us denote by v the solution of the following problem:
where
Now we use the identity
in order to deduce that
By the Euler–Lagrange equation for v we deduce that the sum of the last two integrals in the previous identity is zero being also
Now we prove that u is an ω-minimizer of
(3.17)
Estimate of
(3.18)
where we denoted
Merging the last estimate in (3.18) we deduce
(3.20)
Estimate of
(3.21)
If we choose now
where we used the fact that
Reasoning in a similar way as in (3.19), we can apply the higher integrability for v given by Lemma 6 and infer that
Therefore we obtain
Finally, collecting (3.16), (3.17), (3.20) and (3.22), if we choose
for some constant
Hence we may estimate, using (3.23) and (3.24),
for some constant
We are left with case (iii). Let H be the half-space from our assumption and let us denote accordingly
Let us denote by
where
Let us point out that
Therefore we are in a position to apply Lemma 7 to the function
In addition, using the ellipticity condition of
One more time we can prove that u is an ω-minimizer of
We can estimate the differences
for some constant
4 Energy density estimates
This section is devoted to proving a lower bound estimate for the functional
Scaling.
Let
By direct computations we deduce that the couple
Lemma 9.
Let
for some positive constant
Proof.
Let
Note that, since
If we choose as a representative of E the set of points of density one, we get by Fubini’s theorem that
Combining these inequalities, we can choose
Now we set
If we choose
Then getting rid of the common terms, we obtain
Now if we choose
We choose
Finally, we recall that
From this estimate the result easily follows applying again Lemma 8. ∎
Theorem 5 (Density lower bound).
Let
Moreover,
5 Compactness for sequences of minimizers
In this section we basically follow the path given in [22, Part III]. We start proving a standard compactness result.
Lemma 10 (Compactness).
Let
In addition,
Finally, if we assume also that
for every F such that
Proof.
We start observing that, by the boundedness condition on
and
Now we choose a cut-off function
to test the minimality of
(5.4)
The mismatch term
Now we use the convexity of F and G with respect to the z variable to deduce
where the last two terms in the previous estimate tend to zero as
(5.5)
Finally, we combine (5.4) and (5.5) and pass to the limit as
Letting
and this proves the Λ-minimality of
To prove the strong convergence of
Then from the equi-integrability of
The opposite inequality can be obtained by semicontinuity. Thus we get
From the ellipticity condition in (1.6) we infer, for some
Passing to the limit we obtain
Finally, testing the minimality of
With a usual argument we can deduce
6 Decay of the excess
6.1 Excess and height bound
Now we introduce the usual quantities involved in regularity theory. Given
and the spherical excess of E as
In addition, we define the rescaled Dirichlet integral of u in
The following height bound lemma is a standard step in the proof of regularity.
Lemma 11 (Height bound).
Let
for some
Proof.
The proof of this lemma is almost identical to the one in [22, Theorem 22.8]. Indeed, it follows from the density lower bound (see Theorem 5), the relative isoperimetric inequality and the compactness result proved in the previous section. ∎
Proceeding as in [22] we give the following Lipschitz approximation lemma which is a consequence of the height bound lemma. Its proof follows exactly as in [22, Theorem 23.7].
Theorem 6 (Lipschitz approximation).
Let
then there exists a Lipschitz function
and
where
Finally, we shall need the following reverse Poincaré inequality which can be proved exactly as in the case of Λ-minimizers of the perimeter (see [22, Theorem 24.1]).
Theorem 7 (Reverse Poincaré inequality).
Let
then
for every
6.2 Weak Euler–Lagrange equation
The last ingredient to prove the excess improvement is the following Euler–Lagrange equation that we state for
where
Theorem 8 (Weak Euler–Lagrange equation).
Let
where
Proof.
Let us fix
(6.2)
In order to obtain (6.1), we will divide by t and pass to the upper limit as
We can deal with the second term observing that
(see for instance [16, Theorem 3.2]). In the first bulk term we make the change of variables
Thus we gain
where we set
for
Let us estimate the first of the three terms. By Lipschitz continuity and Young’s inequality we get
Finally, dividing by t and passing to the upper limit as
(6.5)
Passing to the upper limit as
where
7 Excess improvement
Theorem 9 (Excess improvement).
For every
then there exists a positive constant
Proof.
Without loss of generality we may assume that
with some positive constant
Step 1. Thanks to the Lipschitz approximation theorem, for h sufficiently large, there exists a 1-Lipschitz function
We define
and we assume, up to a subsequence, that
We prove that g is harmonic in
for all
Using the Lipschitz continuity of
Therefore, we can prove (7.3). We fix
Therefore, if we replace ϕ by
Decomposing
Since by the second inequality in (7.2) we have
by (7.4) and the area formula, we infer
This proves that g is harmonic.
Step 2. The proof of this step now follows exactly as in [13] using the height bound lemma and the reverse Poincaré inequality. We give it here to be thorough. By the mean value property of harmonic functions, [22, Lemma 25.1], Jensen’s inequality, semicontinuity and the third inequality in (7.2) we deduce that
On one hand, using the area formula, the mean value property, the previous inequality and setting
we have
On the other hand, arguing as in Step 1, we immediately get from the height bound and the first two inequalities in (7.2) that
Hence we conclude that
We claim that the sequence
Therefore, applying the reverse Poincaré inequality and (7.5), we have for h large that
which is a contradiction if we choose
8 Proof of the main theorem
The proof works exactly as in [13]. We give here some details to emphasize the dependence of the constant ε appearing in the statement of Theorem 1 from the structural data of the functional. The proof is divided into four steps.
Step 1.
We show that for every
where
Thanks to the energy upper bound (Theorem 3) and the compactness lemma (Lemma 10) we may assume that
where
Step 2.
Let
Fix
Case 1:
Case 2:
We conclude that choosing
Step 3.
Fix
Note that
In particular,
for some constant
The previous estimate first implies that
Step 4.
Finally, we are in a position to prove that there exists
Being the argument rather standard,
Setting
The conclusion follows as in [13] (see also [6] and [7]) showing that
Acknowledgements
The authors wish to thank the reviewers for the constructive comments and remarks. 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).
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Articles in the same Issue
- Frontmatter
- Harnack inequality for parabolic equations with coefficients depending on time
- Morse theory and the calculus of variations
- Relaxation of functionals with linear growth: Interactions of emerging measures and free discontinuities
- Fractional Poincaré and localized Hardy inequalities on metric spaces
- Higher order Ambrosio–Tortorelli scheme with non-negative spatially dependent parameters
- The notions of inertial balanced viscosity and inertial virtual viscosity solution for rate-independent systems
- Higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions
- On some variational problems involving capacity, torsional rigidity, perimeter and measure
- Sub-elliptic boundary value problems in flag domains
- On master test plans for the space of BV functions
- Regularity results for an optimal design problem with lower order terms
- On the behavior of the first eigenvalue of the p-Laplacian with Robin boundary conditions as p goes to 1
