Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
-
Hong Tian
Abstract
This article is devoted to a global Calderón-Zygmund estimate in the framework of Lorentz spaces for the
1 Introduction
Throughout the article, we assume that
where
for all
holds for any
where
The main aim of this article is to prove a global regularity in the framework of Lorentz spaces for the
where the Lorentz spaces
It is a well-known fact that the uniform ellipticity and boundedness (1.2) is not enough to ensure the validity of the Calderón-Zygmund type estimate (1.5) if one does not impose more regularity assumptions on
where
Assumption 1.1
We say that
there exists a coordinate system depending on
while, for any
for some
and
where
The domain
for any
Definition 1.2
Let
For
We observe that if
Let us now state our research motivation under consideration. It is an important observation that for the linear case of
Some recent studies of problem (1.8) are recalled in order. Byun and Ryu [12] established a global Orlicz estimate of higher-order elliptic and parabolic equations on nonsmooth domains, and they got that
As we know, Iwaniec [32] first proved a local nonlinear Calderón-Zygmund estimate for the equation of
Note that the research on regularity results of higher-order elliptic and parabolic problems is substantially rare in comparison to the second-order cases. Giaquinta and Modica in [25] showed some regularity results for higher-order nonlinear elliptic systems. Campanato and Cannarsa in [13] established a local
We are now in a position to summarize the main result of this article.
Theorem 1.3
For
where the constant
It would be remarked that the restriction for
A brief outline of this article is given as follows: in Section 2, we introduce some preliminary lemmas; Section 3 is mainly devoted to local interior and boundary estimates of the
2 Some technical lemmas
In this section, we collect some preliminary results that will be used in the following context. Throughout the article, we denote by
Lemma 2.1
For any given
If
If
while
There holds a continuous embedding
Note that the boundary of Reifenberg flat domain is locally the graph of a Lipschitz continuous function with a small Lipschitz constant, which ensures the Sobolev embedding theorem and interpolation due to the validity of the Sobolev extension theorem. Let
Lemma 2.2
Let
In the next context, we will intensively use the difference quotients to approximate the weak derivative of the solution. Therefore, it is necessary to recall the following relationship between the difference quotient and the weak derivative, see [24, Theorem 3 in Chapter 5]. For any locally summable function
with
Lemma 2.3
Let
where
Let
for all
Next, it is useful to recall the following two basic inequalities regarding general elliptic operator of
Lemma 2.4
Let the coefficients
for any
Lemma 2.5
Let the coefficients
and
where
It is necessary for the proof of main theorem to recall the following two useful technical lemmas. The first inequality is actually a variant of the classical Hardy’s inequality, whose proof can be found in [6, Lemma 3.4].
Lemma 2.6
Let g be a nonnegative measurable function defined on
then for any
The second lemma is a useful inequality to our main proof, which really encodes the inclusion property with respect to the second exponent of the Lorentz spaces, see [6, Lemma 3.5].
Lemma 2.7
Let
with the constant
with the constant
Finally, the following iterating lemma is mainly used in our main proof, which is referred to [26].
Lemma 2.8
Let
where the constants
3 Local interior and boundary estimates
This section is devoted to some auxiliary estimates by comparing the solution with some reference problems. First, we prove a boundary a priori estimate, which is a crucial ingredient in the proof for a higher integrability of problem (1.1). Note that a similar result can be found in [10, Lemma 6] for problem (1.1) in the interior setting. Here, we only focus our
Lemma 3.1
(Caccioppoli inequality) Let
where
Proof
Without loss of generality, we let
Now, we define
Leibniz’s rule yields that
where
Now we are to estimate
Estimate of I: using the uniform ellipticity equations (1.2) and (3.2), we obtain
Estimates of
for any
for any
for some constant
where
for any
for any
In what follows, we are to show a higher integrability of the
Lemma 3.2
(Reverse Hölder inequality) Assume that
where
Proof
We also let
For
see [2, Theorem 5.4 in Chapter V], where
which implies that
Then, we attain the desired inequality (3.6) by using the Gehring-Giaquinta-Modica lemma, see Proposition 1.1 in [26, p. 122].□
For
and consider the localized problem
under the assumption of
We are now to give various interior comparison estimates by comparing the local original problem with a few of reference problems. Let
with
with
Unlike the
Lemma 3.3
Let
where
In the following lemmas, we set
as the Sobolev conjugate of an arbitrary
Lemma 3.4
Let
moreover, there holds the following estimate:
where
We are next to give the following perturbation estimate by comparing the solutions between the original problem and the limiting problem (3.12). More precisely, we have
Lemma 3.5
Let
Proof
Now, we take
In the following, we part the estimates of equation (3.16) into two cases of
If
for any constant
If
for any
Therefore, putting the two cases into equation (3.16) and using the standard
for any
On the other hand, we take
By Lemma 2.4 and Hölder’s inequality, we have
with any
For the estimate of LHS of equation (3.20), we use a similar argument as in the proofs of (3.17) and (3.18) to obtain
for any
Let us put the above estimates into equation (3.20), and using the standard
Finally, putting equations (3.19) and (3.21) together shows that
for any
In what follows, we are devoted to the relevant estimates in the neighborhood at a point on the flat boundary. Here, we adapt an argument from Skrypnik’s monograph [46] to estimate the normal derivatives controlled by the horizontal derivatives, which is realized by the difference quotient. To this end, we choose a suitable test function consisting of the difference quotient involved the higher derivatives of the solution
for any
where
Lemma 3.6
Let
where
Proof
We divide the proof into two aspects: one is first to estimate the horizontal derivatives
Estimate of the horizontal direction: we select a smooth cut-off function
Let us first confine a small
We use equation (3.24) as a test function, together with the integration by part for the difference quotients to obtain
Note that
We set
and then obtain
with
and
for
Estimate of
Estimate of
If
for any
where
where
for each
where
If
To estimate the second term in the last inequality above, we use an estimate of the difference quotient shown in Lemma 2.3, Poincaré’s inequality, and Young’s inequality with two exponents
for any
Putting the two estimates of
Finally, using Lemma 2.5, we see that
Hence, we put the estimates (3.32) and (3.31) together to yield
This means that the sequence
for each
Estimate of the normal direction: it only suffices to prove
For this, we first need a zero-extension of
Recalling the cut-off function
where
Here,
Using the integration by part for the difference quotients and Leibniz’s rule, it yields that
Let us set
and then, we obtain
Using the uniform ellipticity and boundedness (1.2), we obtain that
and
for
which yields that for any
Putting it into equation (3.37) and again using Leibniz’s rule, we write
where
Estimate of
Estimate of
If
Note that
Putting the above two estimates together, we conclude
If
Thanks to the ellipticity (1.2), the definition of cut-off function (3.23), and Young’s inequality with exponents
To estimate the last two terms for the above inequality, we use the standard estimate of difference quotient as in Lemma 2.3, the Poincaré’s inequality of
Putting the above estimate into the estimate of
Let us now insert the estimates of
It is easy to check that there exists a constant
Thus, this estimate together with equation (3.44) yields
We see that the sequence
Finally let us put equations (3.33) and (3.34) together, then we obtain the desired estimate as shown in Lemma 3.6, which completes the proof.□
With Lemma 3.6 in hand, we can achieve a higher integrability of
Lemma 3.7
Let
where
Proof
With Lemma 3.6 in hand, it suffices to establish a reverse Hölder’s inequality of
To this end, we are first to estimate the horizontal direction, and test equation (3.22) by the following test function:
with a suitable cutting-off function
If
To estimate the second term of the right-hand side above, we use Lemma 2.3 and Poincaré’s inequality of
Now we choose the highest-order coefficient of polynomial
which follows from the Sobolev-Poincaré inequality that
Here, we note that the existence of such polynomial
If
To estimate the second term of the right-hand side above, we also use a similar argument as equations (3.46) and (3.47) so as to attain equation (3.48) for the case of
For the normal direction, we again use the estimate (3.34). Therefore, we put the horizontal estimate (3.49) and the normal estimate (3.34) together to deduce the following reverse Hölder inequality of
Gehring-Giaquinta-Modica lemma (cf. [26, Proposition 1.1]) ensures a higher integrability of
It is easy to check that we make use of the Gagliardo-Nirenberg-Sobolev inequality (cf. [24, Theorem 6]) and Lemma 3.7 to obtain the following conclusion.
Lemma 3.8
Let
with the following estimate:
where
In what follows, we note that the boundary of Reifenberg domain
and
We consider a localized Dirichlet problem
under the assumption of
Let
and let
By the standard
Recalling that
Lemma 3.9
For any
Proof
If
satisfying
and
but
for any weak solution
We note that the Reifenberg domain is locally
Recalling that
for some constant coefficient
Indeed, by the trace theorem
for
Therefore, to attain our claim (3.63), it suffices only to show that
Using Lemma 2.4, we have
for every
Taking
which implies that
Therefore, this combined with equation (3.66) yields
Using equations (3.59), (3.61), (3.62), and (3.64) and taking the limit
Moreover, we also choose
which implies that
Putting the above equality into equation (3.67), we have
Therefore, it yields
Next, we choose
Taking
for every
for every
Finally, it follows from the weak lower semi-continuity of the semi-norm for
The claim (3.63) combined with equation (3.69) leads to a contradiction to equation (3.60) by taking
To make a comparison of the solution
Lemma 3.10
Let
where
Proof
First, we select a smooth cutting-off function
Recall that
in
and take
We are now in a position to estimate the terms LHS and
where
To estimate LHS: we first note that
Using a similar argument to the proofs of equations (3.17) and (3.18), we obtain
for any
To estimate
Employing Young’s inequality and the coefficient assumptions (1.2), we obtain
for any
Now, we extend
To estimate the low-order term involving
for any
For the term
in
This inequality in combination with the Hölder inequality and the reverse Hölder inequality for the Dirichlet problem (3.55) yields
for some
for any
On the other hand, for the Dirichlet problems (3.52), (3.54), and (3.55), we use the same argument as the proof of Lemma 3.5 to obtain
Putting the above two estimates together, we obtain that
which implies the desired result (3.71) by taking enough small constants
4 Proof of main theorem
In this section, we are to prove the global Lorentz estimate of the
where
and consider the upper-level set
Fix any point
for any
On the other hand, if we confine the range:
Consequently, we conclude that for almost every
Then, we infer the following lemma from the well-known Vitali covering lemma due to the property of
Lemma 4.1
Let
and
We are now in a position to show a suitable decay estimate to each member
Lemma 4.2
Under the same hypotheses as Lemma 4.1, we have
where
Proof
By Lemma 4.1, we immediately conclude one of the following alternatives:
For the first case, we split it into two parts and use the Hölder’s inequality to obtain that
which implies
Here, the constant
Taking
Putting it into equation (4.6) and dividing by
For the second case, by the classical measure theory, we estimate the upper level set as follows:
By taking
Putting equations (4.7) and (4.8) together, it completes the proof of Lemma 4.2.□
With Lemma 4.1 in hand, we can construct a disjoint family
We first focus on the estimates concerning the interior case for
By Lemma 4.1 and the definition of
Recalling that
For the sake of simplicity, let us define the following scaling functions in
Then, it is easy to check that
and it follows from equations (4.9)–(4.11) that
and
We readily check that the Dirichlet problem (4.13) still satisfies the hypotheses of Lemmas 3.4 and 3.5, which implies that there exist small
where
we conclude that
We now consider the estimates in the boundary case for
Let us confine
and
where we make
From Lemma 4.1, the definition of
and
Let us make the scaling functions on
Then,
and by equations (4.18)–(4.21), we conclude
and
In similar way, we also check that the Dirichlet problem (4.23) still satisfies the hypotheses of Lemmas 3.8 and 3.10, which implies that there exist small
where
we conclude that
Furthermore, using the expression of domain relation (4.19), we obtain
In short, combining the interior estimates (4.16) with the boundary estimates (4.27), we write
Lemma 4.3
Let
where the parameter
Proof
By Lemma 4.1 and the fact that
For the interior setting: we first note that
where
For the boundary setting: we carry out the same procedure as in equation (4.31) with the boundary estimates (4.27) instead of the interior estimates (4.16) to discover that
Inserting equations (4.31) and (4.32) into equation (4.30), we obtain
Furthermore, using Lemma 4.2 yields
for all
Now we are ready to give the proof of Theorem 1.3.
Proof of Theorem 1.3
Case 1:
Recalling the definition of the upper-level set
Let
To estimate
This combined with the standard
Recalling
To estimate
For the estimate
For the estimate
If
If
Again using a simple change of variable and Hardy’s inequality as Lemma 2.6, we obtain
Putting above two estimates
Combining the estimates
Since
Thus, taking
Furthermore, we take
Therefore, it follows from equation (4.36) that
Case 2:
Also, using the definition of the upper-level set
Let
To estimate
This in combination with the standard
Recalling that
To estimate
For the estimate
For the estimate
Putting
Combining the estimates
Once we obtain the above estimation, the remainder is similar to that of the case
Acknowledgements
The authors would like to thank the anonymous referees for their valuable comments and suggestions that help to improve and clarify the article greatly.
-
Funding information: H. Tian was supported by the National Natural Science Foundation of China (Grant No. 11901429), and S. Zheng was supported by the National Natural Science Foundation of China (Grant Nos 12071021 and 12271021).
-
Author contributions: Both authors contributed equally and significantly to this article.
-
Conflict of interest: The authors state that there is no conflict of interest.
-
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
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- 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
-
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”
Articles in the same Issue
- Regular Articles
- Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
- Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation
- Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
- 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
-
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”
