Abstract
We mainly discuss superquadratic minimization problems for splitting-type variational integrals on a bounded Lipschitz domain \(\Omega \subset {\mathbb {R}}^2\) and prove higher integrability of the gradient up to the boundary by incorporating an appropriate weight-function measuring the distance of the solution to the boundary data. As a corollary, the local Hölder coefficient with respect to some improved Hölder continuity is quantified in terms of the function \({\text {dist}}(\cdot ,\partial \Omega )\).The results are extended to anisotropic problems without splitting structure under natural growth and ellipticity conditions. In both cases we argue with variants of Caccioppoli’s inequality involving small weights.
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1 Introduction
In the main part of our note we consider splitting-type energies as a particular class of variational integrals
defined for functions u: \({\mathbb {R}}^n \supset \Omega \rightarrow {\mathbb {R}}\). In the splitting case, the energy density f: \({\mathbb {R}}^n \rightarrow {\mathbb {R}}\) admits an additive decomposition into different parts depending on the various first partial derivatives of the admissible functions u, for example, we can consider the case
with functions \(f_i\): \({\mathbb {R}}\rightarrow {\mathbb {R}}\), \(i=1\), ..., n, of possibly different growth rates.
This also leads us to the more general category of variational problems with non-standard growth, where in the simplest case one studies an energy functional J defined in equation (1) with convex density f being bounded from above and from below by different powers of \(|\nabla u|\).
For an overview of the various aspects of variational problems with non-standard growth including aspects as existence and regularity we refer the interested reader, e.g., to [1,2,3,4,5,6,7,8] and to the recent paper [9] together with the references quoted therein.
Regarding the question of interior regularity of J-minimizers in the non-standard setting, the above references provide rather complete answers, while the problem of boundary regularity for the solution of
with sufficiently smooth boundary datum \(u_0\) seems to be not so well investigated.
We mention the contribution [10], where the higher integrability of the minimizer u up to the boundary is established for densities f being in some sense close to the splitting-class defined in (2). A precise formulation of the assumptions concerning the density f is given in inequality (2.1) of the paper [10], and we also like to stress that a survey of the known global regularity results is provided in the introduction of this reference.
Even more recently the contribution of Koch [11] addresses the global higher integrability of the gradient of solutions of variational problems with (p, q)-growth, which to our knowledge is the first result to improve global integrabilty of the gradient allowing full anisotropy with different growth rates with respect to different partial derivatives. The boundary data are supposed to belong to some fractional order spaces (see, e.g., [12] or [13]) and roughly speaking are handled like an additional x-dependence following ideas as outlined in, e.g., [14] for the interior situation. Hence, the admissible energy densities correspond to the examples presented in [14]. On the other hand the case of vectorial functions u: \({\mathbb {R}}^n \supset \Omega \rightarrow {\mathbb {R}}^m\), \(n\ge 2\), \(m \ge 1\), as well as the subquadratic situation are included.
In our note we first discuss the variational problem (3) on a bounded Lipschitz domain \(\Omega \subset {\mathbb {R}}^2\) with energy functional J defined in equation (1) under the following assumptions on the data: for \(i=1\), 2 let \(f_i \in C^2({\mathbb {R}})\) satisfy
with constants \(a_i\), \(A_i >0\) and exponents
We define the energy density f: \({\mathbb {R}}^2 \rightarrow {\mathbb {R}}\) according to (recall the “splitting condition” (2))
Note that (4) yields the anisotropic ellipticity condition
valid for Z, \(Y \in {\mathbb {R}}^2\) with positive constants \(c_1\), \(c_2\) and for the choice of exponents \(p=2\) and \(q:= \max \{q_1,q_2\}\). In fact, (7) is a direct consequence of (5) and the formula
Finally, we fix a functiom
remarking that actually \(u_0 \in W^{1,t}(\Omega )\) with exponent t being sufficiently large (depending on \(q_1\) and \(q_2\)) is needed in our calculations. For a definition of the Sobolev spaces \(W^{k,r}(\Omega )\) and their local variants we refer to [12].
The function \(u_0\) acts as a prescribed boundary datum in the minimization problem
Now we state our result in the splitting case by the way establishing global higher integrability without relating \(q_1\) and \(q_2\) and under quite weak assumptions on \(u_0\) which means that we do not have to impose additional hypotheses concerning the regularity of the trace of \(u_0\) in the sense that \({u_0}_{|\partial \Omega }\) belongs to some fractional Sobolev space. The price we have to pay is a weight function which has to be incorporated.
Theorem 1.1
Let the assumptions (4)–(6) and (9) hold. Moreover, let \(u \in u_0 +W^{1,1}_0(\Omega )\) denote the unique solution of problem (10). Then it holds
for any choice of the exponent t such that
with a finite lower bound \(T(q_1,q_2) \ge 1\) depending on the growth rates \(q_1\), \(q_2\).
Remark 1.1
An admissible choice for the quantity \(T(q_1,q_2)\) from (12) is
Here and in the following we let
In the case \(q_i >5\), \(i=1\), 2, we can replace (13) by the expression
which means that in the case of higher initial integrability we can decrease the exponent t in (11) leading to a stronger result. We also note that for the choice \(q_i > 5\), \(i=1\), 2, the condition (9) can be replaced by the requirement \(u_0 \in W^{1,{q_{\min }}}(\Omega )\) together with \(\partial _i u_0 \in L^{q_i}(\Omega )\), \(i=1\), 2.
Remark 1.2
The existence and the uniqueness of a solution u to problem (10) easily follow from the growth properties of f and its strict convexity combined with our assumption (9) on the boundary datum \(u_0\). Obviously the minimizer satisfies \(\partial _i u\in L^{q_i}(\Omega )\), \(i=1\), 2, and from Theorem 1.1 ii) in [8] we deduce interior Hölder continuity of the first partial derivatives. This in turn implies that u is in the space \(W^{2,2}_{{\text {loc}}}(\Omega )\).
Remark 1.3
Of course we expect that our theorem can be extended to the case \(n>2\), i.e. to densities
with functions \(f_i\) satisfying (4) and (5) being replaced by \(q_i > n\), \(i=1\), ..., n. We restrict our considerations in the splitting case to the 2D-situation for two reasons: first, a generalization to the case \(n \ge 2\) would be accompanied by a tremendous technical effort without promising a deeper insight. Secondly, the interior regularity results of [8] as the starting point for our arguments would have to be generalized to the full splitting case in n dimensions which goes beyond the aim of this note.
Remark 1.4
Inequality (11) can be seen as weighted global higher integrability result for \(\nabla u\). In fact, on account of (5), u is Hölder continuous up to the boundary, which means \(|u(x) - u_0(x)| \approx {\text {dist}}^\alpha (x,\partial \Omega )\) for points x near \(\partial \Omega\). Thus, in this vague sense, Theorem 1.1 yields \(|\partial _i u|^{t_i} {\text {dist}}^\beta (\cdot , \partial \Omega ) \in L^1(\Omega )\) for a certain range of exponents \(\beta >0\) and \(t_i > q_i\).
Another interpretation will be given in Corollary 1.1 on the behaviour of a suitable Hölder coefficient.
In the second part of our paper we drop the splitting condition (6) and look at densities f with anisotropic (p,q)-growth in the sense of (7) and for a moment we still restrict our considerations to the two-dimensional case. Then the arguments of the first part can be carried over provided that p and q are not too far apart.
Theorem 1.2
Assume that f: \({\mathbb {R}}^2 \rightarrow {\mathbb {R}}\) is of class \(C^2({\mathbb {R}}^2)\) satisfying (7) with exponents \(2 < p \le q\) such that
Let (9) hold for the boundary datum \(u_0\) and let \(u \in u_0 + W^{1,1}_0(\Omega )\) denote the unique solution of problem (10). Then we have (\(\Gamma := 1 + |\nabla u|^2\))
for any exponents \(\kappa\), s such that
and
Remark 1.5
Since f is strictly convex and since we have (9), existence and uniqueness of a solution u to problem (10) are immediate. Note that (15) implies (4.2) in Marcellini’s paper [15], hence \(u \in W^{1,q}_{{\text {loc}}}(\Omega )\) on account of Theorem 4.1 in [15]. Quoting Corollary 2.2 from this reference we deduce \(u \in C^{1,\alpha }(\Omega )\) and \(u \in W^{2,2}_{{\text {loc}}}(\Omega )\).
In our final theorem we incorporate two new features: we modify the idea of the proofs of the previous results by starting with a more subtle inequality being applied to terms on the left-hand sides of our foregoing calculations by the way extending the range of admissible exponents p and q. Moreover, we include the general case of n-dimensions, \(n \ge 2\). This can be done without technical efforts comparable to the ones which are needed to generalize the 2D splitting case.
Now the precise formulation of our last result is as follows:
Theorem 1.3
Consider the variational problem (10) on a bounded Lipschitz domain \(\Omega \subset {\mathbb {R}}^n\) now just assuming (p,q)-growth of the \(C^2\)-density f: \({\mathbb {R}}^n \rightarrow {\mathbb {R}}\) in the sense of (7) without splitting structure. Let the boundary datum \(u_0\) satisfy (9). Suppose that
By assumption (18) we may choose a real number \(\kappa > 1\) sufficiently large such that
Moreover, let
Increasing \(\kappa\), if necessary, we suppose in addition that
and that we have the inequalities (103) below. Then it holds
for a finite constant c depending on n, p, q, \(\kappa\) and \({\overline{s}}\).
Remark 1.6
Passing to the limit \(\kappa \rightarrow \infty\), (20) becomes
Remark 1.7
We note that Remark 1.5 extends to the setting studied in Theorem 1.3, since now condition (4.2) from [15] is a consequence of the more restrictive inequality (18).
Our paper is organized as follows: in Section 2 we present the proof of Theorem 1.1 for a given Lipschitz boundary datum \(u_0\) (recall (9)) and just supposing (5). In the case \(q_1\), \(q_2 > 5\) the lower bound \(T(q_1,q_2)\) occurring in (12) looks much nicer and (9) can be considerably weakened. We sketch some arguments in Section 3. The two-dimensional non-splitting case, i.e. the setting described in Theorem 1.2, is discussed in Section 4 and in Section 5 we discuss the higher dimensional situation of Theorem 1.3.
During the proofs of our results we make essential use of Caccioppoli-type inequalities. As a main new feature these inequalities involve “small weights”, for instance weight functions like \((1+|\partial _i u|)^\alpha\), \(i=1\), 2, with a certain range of negative exponents \(\alpha\). A rather general analysis of these tools is presented in Section 6.
Before we get into all these technical details we finish the introduction by adding some extra comments concerning the interpretation of our results. For simplicity we restrict ourselves to the setting described in Theorem 1.1 and denote by u the unique solution of problem (10) under the hypothesis of Theorem 1.1.
We start with a local observation: by adding a distance function if necessary, we find an appropriate representative of the boundary data respecting a lower bound for \(|u-u_0|\) in a neighbourhood of a fixed \(x_0 \in \Omega\).
Proposition 1.1
Fix \(x_0 \in \Omega\). Then there exists a neighbourhood \(U=U(x_0)\Subset \Omega\) of \(x_0\) such that we may assume that for all \(x \in U\)
where
denotes the distance function to the boundary \(\partial \Omega\).
Proof of Proposition 1.1
The distance function d is discussed, e.g., in [16], Appendix 14.6. In particular, if \(\Omega\) is a domain in \({\mathbb {R}}^n\) with non-empty boundary, then by (14.91) of [16] d is uniformly Lipschitz continuous.
Let us suppose w.l.o.g. that \({\hat{u}}(x_0) \ge 0\). If \({\hat{u}}(x_0) < d(x_0)\), then we choose \({\tilde{u}}_0 := u_0-d\) as admissible Lipschitz boundary datum with \(u_0 = {\tilde{u}}_0\) on \(\partial \Omega\). Hence both \(u_0\) and \({\tilde{u}}_0\) produce the same minimizer u of the Dirichlet problem (10) and by definition we have
The considerations of our note completely remain unchanged if we replace \(u_0\) by \({\tilde{u}}_0\). Hence we have w.l.o.g. that \({\hat{u}}(x_0) \ge d(x_0)\). Proposition 1.1 then follows from the continuity of u and \(u_0\). \(\square\)
Let us fix \(x_0\in \Omega\), and define U, \({\hat{u}}\) as in Proposition 1.1. Moreover we let h: \(\Omega \rightarrow {\mathbb {R}}\), \(h:= |{\hat{u}}|^{\frac{4}{3} \frac{\kappa }{{q_{\min }}}+1}\), hence
Theorem 1.1 gives
Letting \(\zeta = \frac{4}{3} \frac{\kappa }{{q_{\min }}}+1\), the imbedding into Hölder spaces yields \(|{\hat{u}}|^{\zeta } \in C^{0,\mu }({\overline{\Omega }})\) with \(\mu = 1- \frac{4}{3{q_{\min }}}\).
Now we consider the Taylor expansion of the function g: \({\mathbb {R}}^+ \rightarrow {\mathbb {R}}\), \(w \mapsto w^{\zeta }\) around a fixed \({\tilde{w}} >0\): for w sufficiently close to \({\tilde{w}}\) we have
Suppose w.l.o.g. that \({\hat{u}} > 0\) in U. Inserting \({\tilde{w}} = {\hat{u}}(x_0)\) and \(w = {\hat{u}}(x)\), x sufficiently close to \(x_0\), in the Taylor expansion we obtain using Proposition 1.1
Since \(|{\hat{u}}|^\zeta\) is of class \(C^{0,\mu }({\overline{\Omega }})\) and since according to (9) \(u_0\) is Lipschitz, by (22) we find a constant \(c >0\) not depending on \(x_0\) such that
Changing the roles of x and \(x_0\) and following the arguments leading to (23), we find a constant \(c >0\) not depending on \(x_0\) such that for all x sufficiently close to \(x_0\) we have
Summarizing the results we obtain the following corollary to Theorem 1.1.
Corollary 1.1
Under the assumptions of Theorem 1.1 we let
If \(x_0 \in \Omega\) is fixed, then for every sufficiently small neighbourhood \(U = U(x_0)\Subset \Omega\) of \(x_0\) the Hölder coefficient of u (see [16] for notation) satisfies
where the constant c is not depending on \(x_0\).
2 Proof of Theorem 1.1
We proceed in several steps assuming from now on the validity of the hypotheses (4), (5), (6) and (9). Let u denote the unique solution of problem (10) and recall the (interior) regularity properties of u stated in Remark 1.2. We start with the following elementary observation:
Proposition 2.1
Consider numbers \(\kappa\), \(s_1\), \(s_2 \ge 1\). Then there is a finite constant c such that for \(i=1\), 2 and any \(\eta \in C^1_0(\Omega )\)
Proof of Proposition 2.1
Let \(v:= \Gamma _i^{s_i/2} |\eta |^{\kappa }\) and observe that due to \(u \in W^{2,2}_{{\text {loc}}} \cap W^{1,\infty }_{{\text {loc}}}(\Omega )\) the function v is in the space \(W^{1,1}_{0}(\Omega )\), hence the Sobolev-Poincaré inequality implies
and inequality (25) directly follows by observing that \(\big |\nabla |\eta |\big | \le |\nabla \eta |\). \(\square\)
Next we replace \(\eta\) by a sequence \(\eta _m\), \(m\in {\mathbb {N}}\), being defined through
with \(\varphi _m \in C^1_0(\Omega )\), \(0 \le \varphi _m \le 1\),
Note that due to (5) and (9) inequality (25) extends to \(\eta _m\), moreover, according to [16], Theorem 7.17, we have
for points \(x\in \Omega\) at distance \(\le \rho\) to \(\partial \Omega\), provided \(\rho\) is sufficiently small.
The quantity \((II)_i\) defined in (25) with respect to the choice \(\eta = \eta _m\) behaves as follows:
Proposition 2.2
Let \(\delta := {q_{\min }}-2\) and consider numbers \(s_i\), \(\kappa \ge 1\), \(i=1\), 2, such that
Then it holds for \((II)_i = (II_m)_i\), \(i=1\), 2, defined in (25) with respect to the function \(\eta _m\) from (26)
c denoting a finite constant being uniform in m.
Proof of Proposition 2.2
By Hölder’s inequality we have
and for the second integral on the r.h.s. we observe (recall (26) and (27), (28))
Since \(\Omega\) is Lipschitz we observe that \(|{\text {spt}} \nabla \varphi _m| \le c/m\) and obtain
The choice (29) of \({{\hat{\kappa }}}\) shows that the right-hand side of (32) is bounded. Letting \(\Gamma := 1 + |\nabla u|^2\) we arrive at
where in the final step we have used the definition of \(\delta\). Now (30) clearly is a consequence of (9) (in fact, \(u_0 \in W^{1,{q_{\min }}}(\Omega )\) would be sufficient). \(\square\)
In the following we will discuss the quantity \((III)_i\) defined in (30) under the assumptions of Proposition 2.2.
We have by Young’s inequality for any \(\varepsilon > 0\)
with exponent
\([\dots ]^*\) denoting the exponent conjugate to \(\frac{s_i}{s_i - \delta /2}\). Note that \(\vartheta \ge 0\) provided we additionally assume that \(\kappa \ge 1\) satisfies the inequality
Inserting the above estimate for \((III)_i\) into (30), we find
and if we choose \(\varepsilon\) sufficiently small, we see that (25) yields the following result:
Proposition 2.3
Let \(s_i\), \(\kappa \ge 1\) satisfy (29) and (33). Then there exists a constant c being independent of m such that (\(i=1\), 2)
In order to prove Theorem 1.1 it remains to discuss the quantity \((I) = (I)_i\) for \(i=1\), 2, i.e. now second derivatives of u have to be handled in an appropriate way. It holds
and in the same manner
We have
where the last estimate follows from Hölder’s inequality and \(\alpha _i\), \(i=1\), 2, denote real numbers such that for the moment
Note that the condition (35) guarantees the validity of
for a finite constant c. Recalling (4) we see that (36) yields the bound
again for \(i=1\), 2.
Let us look at the quantities \({\tilde{T}}_i\): we have by Hölder’s inequality
In order to benefit from the inequality (38), we replace (35) by the stronger bound
together with the requirement
Here we note that (39) together with (40) yields (35). We then obtain on account of Young’s inequality
with exponent
where we used (39), (40) and (40), (5), respectively, for the last inequalities. Thus (38) reduces to
In the same spirit it follows
Let us return to (34): we have by (37), (41) and (42) and by (8)
The remaining integrals are handled with the help of Proposition 6.1 (compare also inequality (4.6) from [17]) replacing l by \(\kappa\) and \(\eta\) by \(|\eta _m|\), respectively. We note that the proof of Proposition 6.1 obviously remains valid with these replacements. We emphasize that (40) is an essential assumption to apply Proposition 6.1. We get
Finally we let \(s_i = \frac{3}{4}q_i\), \(i=1\), 2, which is the optimal choice with respect to (39). We note that with this choice (33) follows from
and (44) is valid if \(\kappa\) is chosen according to assumption (12) from Theorem 1.1 with \(T(q_1,q_2)\) defined in (13).
We recall the definition (26) of \(\eta _m\) and the gradient bound for \(\varphi _m\) stated in (27). This yields for integrand of the first integral on the right-hand side of (45):
We quote inequality (28) and recall that \(\nabla \varphi _m\) has support in the set \(\{x\in \Omega :\, {\text {dist}}(x,\partial \Omega )\le 1/m\}\) satisfying \(|{\text {spt}}\nabla \varphi _m|\le c/m\), hence
where (recall (40))
Thus the second integral on the right-hand side is bounded if, with \(\alpha _i\) sufficiently close to \(-1/2\), we have
Assuming (46) we arrive at
for any \(\alpha _i\) sufficiently close to \(-1/2\). Assuming this we next let
and apply Young’s inequality in an obvious way to get
where the first term can be absorbed in the left-hand side of (43), while the second one bounded under the assumption
Here we used the fact that for \(q > 2\) the function \(q/(q-2)\) is a decreasing function. Altogether we have shown that for exponents \(\alpha _i\) close to \(-1/2\) the first integral on the right-hand side of (45) splits into two parts, where the first one can be absorbed in the left-hand side of (45) and the second one stays bounded. During our calculations we evidently used (9), however (9) can be replaced by weaker integrability assumtions concerning \(\partial _i u_0\). We leave the details to the reader.
Let us finally consider the “mixed terms” on the right-hand side of (45). We first observe the inequality
Considering the limit case \(\alpha _1 =-1/2\), the first integral on the right-hand side of (49) basically is of the form
and this integral directly results from an application of Caccioppoli’s inequality. If we like to show an integrability result for \(\Gamma _2^{t_2}\) with some power \(t_2 > q_2/2\), then the idea is to apply Young’s inequality choosing
leading to the quantities \(\Gamma _2^{t_2}\) and \(\Gamma _1^{\frac{t_2}{2t_2 - q_2}}\). If \(t_1 =t_1(q_1) > q_1/2\) denotes the desired integrability exponent for \(\Gamma _1\), then this requires the bound
and of course we need the same condition changing the roles of \(t_1\) and \(t_2\). With the symmetric Ansatz \(t_i = \theta q_i\) for some \(\theta > 1/2\) we are immediately led to \(t_i = \frac{3}{4} q_i\), \(i=1\), 2, which again motivates our choice of \(s_i\).
More precisely: discussing the first integral on the right-hand side of (49) we choose \(\beta _1 = 3/2\) with conjugate exponent \(\beta _1^* =3\) and obtain
Here the first integral is absorbed in the left-hand side of (43) and since \(\alpha _1\) is chosen sufficiently close to \(-1/2\), the second integral is bounded provided that we suppose in addition
If \(q_1 < 3\), using
we are led to (\({\tilde{\varepsilon }} \ll \varepsilon\))
where the first integral is absorbed in the right-hand side of (43) and if \(\alpha _1\) sufficiently close to \(-1/2\) we now suppose
Note that the condition (51) is a consequence of (48).
It remains to discuss the second integral on the right-hand side of (49). Using Young’s inequality with
we obtain
where again the first integral is absorbed in the left-hand side of (43). Considering the second integral we choose
This gives (\({\tilde{\varepsilon }} \ll \varepsilon\))
As usual the first integral on the right-hand side of (53) is absorbed in the left-hand side of (43) and we calculate
We obtain recalling (27) and (28)
For \(\alpha _1\) sufficiently close to \(-1/2\) this leads to the requirement
With \(q_1>2\) fixed and for \(q_2 > 2\) we consider the function \(g(q_2) = \frac{6q_1q_2}{q_2(q_1-2) + 4q_1}\). We have
hence g is an increasing function and
Thus,
implies the validity of (56).
Summarizing the conditions imposed on \(\kappa\) during our calculations, i.e. recalling the bounds (29), (44), (46), (48), (57) we are led to the lower bound
for the exponent \(\kappa\). Assuming the validity of (58) and returning to (43) we now have shown that for \(\alpha _i\) sufficiently close to \(-1/2\) the right-hand side of (43) can be splitted into terms which either can be absorbed in the left-hand side of (43) or stay uniformly bounded, hence
for a finite constant c independent of m. Passing to the limit \(m \rightarrow \infty\) in (59) our claim (11) follows. Obviously (12) is a consequence of (58) and the definition of \(T(q_1,q_2)\) stated in (13), which completes the proof of Theorem 1.1 for arbitrary exponents \(q_1\), \(q_2 >2\). \(\square\)
Remark 2.1
Let us add some comments on the behaviour of \(\kappa\), which means that we look at the lower bound for the exponent \(\kappa\) given by the right-hand side of inequality (58).
-
i)
Since \(u-u_0 \in W^{1,{q_{\min }}}(\Omega )\) and thereby \(u-u_0 \in C^{0,\nu }({\overline{\Omega }})\), \(\nu := 1- 2/{q_{\min }}\), the Hölder exponent enters (58), which also corresponds to the natural effect that
-
ii)
\(\kappa \rightarrow \infty\) as \({q_{\min }}\rightarrow 2\).
-
iii)
The ratio \({q_{\max }}/{q_{\min }}\) determines the growth of \(\kappa\).
-
iv)
In the limit \({q_{\max }}= {q_{\min }}\rightarrow \infty\) condition (58) reduces to \(\kappa > 3\).
3 Comments on the case \({q_{\min }}> 5\)
We now choose the sequence \(\eta _m \in C^1_0(\Omega )\) according to
For some elementary properties of anisotropic Sobolev spaces including an appropriate version of this density result we refer, e.g., to [18, 19]). We emphasize that during the following calculations condition (9) can be replaced by the weaker requirement \(\partial _i u_0 \in L^{q_i}(\Omega )\), \(i=1\), 2. Proposition 2.1 obviously holds for \(\eta _m\), and with \(\delta\) as in Proposition 2.2 we now obtain (30) with the choice \({{\hat{\kappa }}}=1\) observing that going through the proof of (30) the quantity \((II)_i\) can be handled as follows: first we note
and then we use
which is a consequence of (60).
We continue with the discussion of \((III)_i\) for the choice \({{\hat{\kappa }}}=1\), and observe that (33) has to be replaced by
Exactly as in Section 2 we obtain the inequalities (43), (45) now being valid for any exponents \(s_i\), \(\alpha _i\), \(\kappa\) such that (\(i=1\), 2)
Still following the lines of Section 2 we let \(s_i = \frac{3}{4} q_i\), \(i=1\), 2 , and replace (61) by
We let
and consider the terms on the right-hand side of (45). Young’s inequality yields (\(0< \varepsilon < 1\))
with
While the \(\varepsilon\)-part of (64) can be absorbed, the remaining integrals are bounded if for \(i=1\), 2
Noting that the function \(t \mapsto t/(t+2)\), \(t \ge 0\), is increasing and by choosing \(\alpha _i\), \(i=1\), 2, sufficiently close to \(-1/2\), we see that these requirements are consequences of the strict inequalities
We thus have assuming (63) and (65)
Let us have closer look at the first integral on the right-hand side of (66). Youngs’s inequality gives
Here the first integral is absorbed and the second is bounded if we have (again \(\alpha _2\) being sufficiently close to \(-1/2\))
The first condition follows from the requirement (65), the second one holds if we assume in addition that \({q_{\min }}> 5\).
In the same way the last term on the right-hand side of (66) is handled and by combining (63), (65) we have shown Theorem 1.1 together with the formula (14). \(\square\)
4 Proof of Theorem 1.2
We proceed along the lines of Section 2 assuming that all the hypothese of Theorem 1.2 are satisfied. In place of (25) we have (\(\Gamma := 1+|\nabla u|^2\))
and (68) holds for any s, \(\kappa \ge 1\) and \(\eta \in C^1_0(\Omega )\). We define the sequence \(\eta _m\) as in (26) observing that (28) holds with \({q_{\min }}\) replaced by p. Letting \(\delta := p-2\), defining \({\hat{\kappa }} := \frac{p-1}{p-2}\) we obtain for
applying obvious modifications in the proof of (30)
Here (II) just denotes the term occurring on the right-hand side of (68) with \(\eta\) being replaced by \(\eta _m\). The quantity (III) is discussed analogously to the term \((III)_i\): in accordance with the calculations presented after the proof of Proposition 2.2 we get for any \(\varepsilon >0\)
wtith exponent \(\vartheta \ge 0\), if we assume that
is satisfied. Choosing \(\varepsilon\) sufficiently small, inserting (71) in (70) and returning to (68) it is shown (compare (35))
Adjusting the calculations presented after inequality (35) to the situation at hand we consider numbers s, \(\alpha\) satisfying
and find recalling (7) and using (74)
where from now on the sum is taken with respect to the index i. Let us assume in addition to (74) that we have
as lower bound for the parameter \(\alpha\). Quoting (107) from Proposition 6.2 (with \(\eta\) being replaced by \(|\eta _m|^\kappa\)) and returning to (73) we find:
On the right-hand side of (76) we use the second inequality from the ellipticity condition (7) to obtain (in analogy to the inequalities (43), (45))
The right-hand side of (77) is discussed following the arguments presented after (45): we first have (recall (28))
where we have defined
Obviously this requires the bound \(\frac{p}{2} \ge \frac{q}{2}+\alpha\), i.e. the condition
in combination with (75) yields the maximal range of anisotropy , since then we can choose \(\alpha\) sufficiently close to \(-1/4\) to guarantee \(p > q+2\alpha\). Moreover, we assume the validity of \(-1 + \gamma ^*[2-2\kappa (p-2)/p]\le 0\). This is true again for \(\alpha\) chosen sufficiently close to \(-1/4\), provided that
Recalling (78) we see that (79) is a consequence of the stronger bound
In accordance with (47) we therefore arrive at
Neglecting the contribution resulting from \(\nabla u_0\) and under the additional hypothesis
which guarantees the validity of \(s > \frac{q}{2}+1+\alpha\) for \(\alpha > -1/4\), we have
Let us add a comment concerning the conditions imposed on s: as remarked after (78) and during the subsequent calculations the parameter \(\alpha\) has to be adjusted in an appropriate way and might become very close to the critical value \(-1/4\). For this reason inequality (74) is replaced by the stronger one (recall (16))
In order to find numbers s satisfying (82) and (85) we need the bound \(q < 2p-2\) being more restrictive for \(p \in (2, 5/2)\) than inequality (78).
Finally we note that the exponent \(2\kappa - 2 \beta ^*\) occurring in the second integral on the right-hand side of (83) is non-negative provided \(\kappa \ge s/(s-[1+\frac{q}{2}+\alpha ])\), and the latter inequality holds for \(\alpha\) sufficiently close to \(-1/4\), if (compare (17))
Assuming (86) our claim follows by inserting (83) into (81) and choosing \(\varepsilon\) sufficiently small. \(\square\)
5 Proof of Theorem 1.3
Let all the assumptions of Theorem 1.3 hold. We define the number
and note that
We also remark that for any fixed \(\alpha\) and for any \(\kappa >1\) the inequality
is equivalent to the requirement \(\alpha > - p/n\).
Lemma 5.1
Fix \(\alpha > - 1/(2n)\), \(\kappa > 1\) with (87) and choose a real number \({\overline{s}}_\alpha\) satisfying
Then we have for any \(\varepsilon >0\) and for \(\eta \in C^{1}_0(\Omega )\)
for a constant being independent on \(\eta\), where s is given by (90).
Proof of Lemma 5.1
We first observe that for
we have after an application of Young’s inequality
for any \({\tilde{\varepsilon }} > 0\). We then estimate using Sobolev’s inequality
Let us first consider \(T_1\). Hölder’s inequality gives
For \(T_{1,2}\) we observe recalling (90)
which is true by the choice (88) of \({\overline{s}}_\alpha\), hence \(T_{1,2}\) is uniformly bounded.
We handle \(T_{1,1}\) with the help of Proposition 6.2 (replacing \(\eta\) by \(|\eta |^\kappa\)):
with constants c being independent of \(\eta\) and it remains to discuss \(T_2\) in (95): we have for \({\hat{\mu }} =2/\gamma\), \(\mu = 2/(2-\gamma )\),
With (91) for \({\tilde{\varepsilon }} < \varepsilon /c\) sufficiently small, (95) and (96) the proof of Lemma 5.1 is completed. \(\square\)
Now we come to the proof of the theorem: we choose \(\eta _m = (u - u_0)\varphi _m\) with \(\varphi _m\) defined after (26). Then Lemma 5.1 yields
Here the first two integrals on the right-hand side can be absorbed in the left-hand side provided that we have (recall that \(u_0\) is Lipschitz)
Also on account of (98) we can handle the remaining integrals on the right-hand side of (97) with the help of Young’s inequality. We obtain
as well as
In (99) we choose \(\beta _1\) such that
with conjugate exponent (recall \(q-p < 2p/n\) on account of (18))
Note that for \(\alpha\) sufficiently close to \(-1/(2n)\) it holds (recall 88)
In (100) we define \(\beta _2\) according to
with conjugate exponent
and choosing \(\alpha\) close to \(-1/(2n)\) we can arrange that (recall 88)
Returning to (99) and (100), respectively, the first integral again is absorbed in the left-hand side of (97). The remaining integrals stay bounded if we have for \(i=1\), 2
Thus we require the condition
which on account of \(p>n\) is satisfied for \(\kappa\) sufficiently large.
It remains to arrange (98) together with the inequality on the right-hand side of (88). Here we first observe
For \(\alpha\) sufficiently close to \(-1/(2n)\), (104) is a consequence of (19). We finally have to discuss
Assumption (21) implies (105). Hence, (97), (99) and (100) prove Theorem 1.3 passing to the limit \(m\rightarrow \infty\). \(\square\)
6 Appendix. Caccioppoli-type inequalities
We prove two Caccioppoli-type inequalities with small weights (i.e. involving powers of \(\Gamma = 1+ |\nabla u|^2\) or of \(\Gamma _i =1+|\partial _i u|^2\), \(i=1\), ..., n, with a certain range of negative exponents), where the first one is the appropriate version in the splitting context.
There is no need to restrict the following considerations to the case \(n=2\). Thus, throughout this appendix, we suppose that \(\Omega \subset {\mathbb {R}}^n\) is a bounded Lipschitz domain and that f: \({\mathbb {R}}^n \rightarrow {\mathbb {R}}\) is of class \(C^2\) satisfying \(D^2f(Z)(Y,Y) > 0\) for all Z, \(Y \in {\mathbb {R}}^n\).
Moreover we suppose that \(u \in W^{2,2}_{{\text {loc}}}(\Omega )\cap C^{1}(\Omega )\) solves the differentiated Euler equation
and for any \(1 \le i \le n\) fixed.
Proposition 6.1
Fix \(l\in {\mathbb {N}}\) and suppose that \(\eta \in C^\infty _0(\Omega )\), \(0 \le \eta \le 1\). Then the inequality
holds for any \(\alpha > - 1/2\) and for any fixed \(1\le i \le n\).
Proof of Proposition 6.1
Suppose that \(-1/2 < \alpha\) and fix \(1 \le i \le n\) (no summation with respect to i). Using approximation arguments we may insert
in the equation (106) with the result
In (108) we have
which gives \(S_1 \le 0\) if \(\alpha \ge 0\). In this case we will just neglect \(S_1\) in the following. In the case \(-1/2< \alpha < 0\) we estimate
Since we have \(2 |\alpha | < 1\) we may absorb \(|S_1|\) in the left-hand side of (108), hence
For \(0 < \varepsilon\) sufficiently small we apply the Cauchy-Schwarz inequality to discuss \(S_2\):
After absorbing the first term in the right-hand side of (108) we have established our claim (107). \(\square\)
Instead of the quantities \(\Gamma _i\) our second inequality (compare [20] for the discussion in two dimensions) involves the full derivative, i.e. we incorporate certain negative powers of \(\Gamma = 1+|\nabla u|^2\). As a consequence we do not obtain the range \(-1/2 < \alpha\) and have to replace this condition by the requirement \(-1/(2n) < \alpha\).
Proposition 6.2
Suppose that \(\eta \in C^{\infty }_{0}\big (\Omega \big )\) and fix some real number \(\alpha\) such that \(- 1/(2n) < \alpha\). Then we have (summation with respect to \(i =1\), ..., n)
where the constant c is not depending on \(\eta\). In particular it holds
Proof
For \(i =1\), ..., n and any \(\eta\) as above we deduce from (106)
where at this stage no summation with respect to the index i is performed. Following [20] we denote the bilinear form \(D^2f(\cdot ,\cdot )\) by \(\langle \cdot ,\cdot \rangle\). The first observation is the inequality
For handling the integrand of the term I from (111) we use the identity
The last term on the right-hand side of (113) is estimated as follows
This, together with (113) gives (recall \(-2\alpha > 0\))
Combining (114) and (112) we get
Returning to (111) and using (115) we get (from now on summation with respect to i)
This finishes the proof by applying the Cauchy-Schwarz inequality. \(\square\)
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Bildhauer, M., Fuchs, M. On the global regularity for minimizers of variational integrals: splitting-type problems in 2D and extensions to the general anisotropic setting. J Elliptic Parabol Equ 8, 853–884 (2022). https://doi.org/10.1007/s41808-022-00179-4
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DOI: https://doi.org/10.1007/s41808-022-00179-4