A Sharp Nonlinear Gagliardo–Nirenberg-Type Estimate and Applications to the Regularity of Elliptic Systems

Q: What is the simplest case of compactly supported smooth functions?

In the simplest case of compactly supported smooth functions on n, it reads∫ n u s+2 dx ≤ C n s u 2BMO ∫ n u s−2 2u 2 dx s ≥ 2 (1.3)A more precise local variant, which forms one of the main results of this paper, is stated below, in Section 3, as Theorem 3.1.

Question

1. What contributions have the authors mentioned in the paper "A sharp nonlinear gagliardo–nirenberg-type estimate and applications to the regularity of elliptic systems" ?

2. What is the name of the paper?

3. What is the simplest way to prove the monotonicity of u?

4. What is the strategy behind the proof of regularity?

Accepted Answer

In this paper, Riviere et al. showed that the BMO norm of a function can be computed in the Hardy space.

Accepted Answer

The seminal paper of Coifman et al. (1993) has triggered numerous new applications of Hardy spaces and the space BMO of functions of bounded mean oscillation to nonlinear partial differential equations.

Accepted Answer

Since second derivatives of u exist in Lp, one can integrate (1.10) against test vectors uxj to obtain∫ u pdiv dx = p∑j k l∫ u p−2 u jxkujxl l xk dx for all ∈ C 0 nThus the monotonicity formula is automatically satisfied, u ∈ BMO, and its local BMO norm is controlled by the scaled energy; see, e.g., Evans (1991) (for p = 2 the computation is identical).

Accepted Answer

As the authors already mentioned in the introduction, the strategy behind the proof of -regularity is quite simple: Theorem 3.1 is used to absorb the integral of u p+2+2 , which appears in the right-hand side of Caccioppoli inequality (2.1), and then Moser iteration works.

Accepted Answer

The assumption that u is a local minimum of energy is crucial: there are examples of harmonic maps u 3 → S2 that are nowhere continuous, see Rivière (1995).

Accepted Answer

For the remaining part of the computations that are necessary to prove Theorem 1.2, it is convenient to choose qmax = p+ 6n, max = 3n.

Accepted Answer

owing to the strength of the inequality (1.3), Theorems 1 and 2 can be applied to arbitrary—not just minimizing—p-harmonic maps in W 2 p.

Accepted Answer

There exists a constant 1 = 1 n N p such that if B ⊂ andmax { u BMO B ( p − ∫ B u p dx )1/p} < 1then u ∈ L B /4 andessmax B /4u ≤ C n N p( p − ∫ B u p dx )1/p(1.7)These two theorems are closely related to results of Duzaar and Mingione (2004) who generalize to the p-harmonic case Simon’s approach to the regularity theory of harmonic maps and prove the p-harmonic approximation lemma.

Accepted Answer

In the simplest case of compactly supported smooth functions on n, it reads∫ n u s+2 dx ≤ C n s u 2BMO ∫ n u s−2 2u 2 dx s ≥ 2 (1.3)A more precise local variant, which forms one of the main results of this paper, is stated below, in Section 3, as Theorem 3.1.

A Sharp Nonlinear Gagliardo–Nirenberg-Type Estimate and Applications to the Regularity of Elliptic Systems

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Most frequently asked questions

1. What contributions have the authors mentioned in the paper "A sharp nonlinear gagliardo–nirenberg-type estimate and applications to the regularity of elliptic systems" ?

2. What is the name of the paper?

3. What is the simplest way to prove the monotonicity of u?

4. What is the strategy behind the proof of regularity?

5. What is the condition for minimizing p-harmonic maps?

6. What is the qmax for the remaining part of the computations that are necessary to prove?

7. Why can't theorems 1 and 2 be applied to p-harm?

8. What is the constant of the p-harmonic system?

9. What is the simplest case of compactly supported smooth functions?

Citations

A Gagliardo-Nirenberg type inequality for rapidly decaying functions

A conditional regularity result for p-harmonic flows

Interpolation réelle des espaces de Sobolev sur les espaces métriques mesurés et applications aux inégalités fonctionnelles

Local and global existence of solutions to a fourth-order parabolic equation modeling kinetic roughening and coarsening in thin films

A Gagliardo–Nirenberg Type Inequality for Rapidly Decaying Functions

References

Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals

H p spaces of several variables

Weakly differentiable functions

On Harnack's theorem for elliptic differential equations†

A regularity theory for harmonic maps

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