Abstract:We extend several well-known tools from the theory of secondorder divergence-form elliptic equations to the case of higher-order equations. These tools are the Caccioppoli inequality, Meyers's reverse Hölder inequality for gradients, and the fundamental solution. Our construction of the fundamental solution may also be of interest in the theory of second-order operators, as we impose no regularity assumptions on our elliptic operator beyond ellipticity and boundedness of coefficients.2010 Mathematics Subject C… Show more
“…The main result of the paper was a construction of the fundamental solution in the case of higher‐order operators. was constructed as an order‐ m antiderivative of the kernel to the operator , the Newton potential for L , defined as follows.…”
Section: Definitionsmentioning
confidence: 99%
“…We will need two additional properties of the Newton potential from . First, we will need the symmetry relation for all and all .…”
Section: Definitionsmentioning
confidence: 99%
“…The main result of may be stated as follows. Theorem Let L be an operator of order 2 m that satisfies the bounds and .…”
Section: Definitionsmentioning
confidence: 99%
“…(The derivative is included in formula because , and so is also defined only up to adding polynomials.) If q and s are small enough, then there is a unique normalization of the derivatives that satisfies the bound or ; in this normalization was found using the Gagliardo–Nirenberg–Sobolev inequality. However, if then itself (and possibly some of its derivatives) are still not well‐defined.…”
Section: Definitionsmentioning
confidence: 99%
“…We begin with some bounds on solutions to elliptic equations. Specifically, we begin with the following higher‐order generalization of the Caccioppoli inequality; in its full generality it was proven in , but the case was proven in and an intriguing version appears in . Lemma Suppose that L is a divergence form elliptic operator associated to coefficients satisfying the ellipticity conditions and .…”
In this paper we establish square‐function estimates on the double and single layer potentials for divergence form elliptic operators, of arbitrary even order 2m, with variable t‐independent coefficients in the upper half‐space. This generalizes known results for variable‐coefficient second‐order operators, and also for constant‐coefficient higher‐order operators.
“…The main result of the paper was a construction of the fundamental solution in the case of higher‐order operators. was constructed as an order‐ m antiderivative of the kernel to the operator , the Newton potential for L , defined as follows.…”
Section: Definitionsmentioning
confidence: 99%
“…We will need two additional properties of the Newton potential from . First, we will need the symmetry relation for all and all .…”
Section: Definitionsmentioning
confidence: 99%
“…The main result of may be stated as follows. Theorem Let L be an operator of order 2 m that satisfies the bounds and .…”
Section: Definitionsmentioning
confidence: 99%
“…(The derivative is included in formula because , and so is also defined only up to adding polynomials.) If q and s are small enough, then there is a unique normalization of the derivatives that satisfies the bound or ; in this normalization was found using the Gagliardo–Nirenberg–Sobolev inequality. However, if then itself (and possibly some of its derivatives) are still not well‐defined.…”
Section: Definitionsmentioning
confidence: 99%
“…We begin with some bounds on solutions to elliptic equations. Specifically, we begin with the following higher‐order generalization of the Caccioppoli inequality; in its full generality it was proven in , but the case was proven in and an intriguing version appears in . Lemma Suppose that L is a divergence form elliptic operator associated to coefficients satisfying the ellipticity conditions and .…”
In this paper we establish square‐function estimates on the double and single layer potentials for divergence form elliptic operators, of arbitrary even order 2m, with variable t‐independent coefficients in the upper half‐space. This generalizes known results for variable‐coefficient second‐order operators, and also for constant‐coefficient higher‐order operators.
Recent years have brought significant advances in the theory of higher order elliptic equations in non-smooth domains. Sharp pointwise estimates on derivatives of polyharmonic functions in arbitrary domains were established, followed by the higher order Wiener test. Certain boundary value problems for higher order operators with variable non-smooth coefficients were addressed, both in divergence form and in composition form, the latter being adapted to the context of Lipschitz domains. These developments brought new estimates on the fundamental solutions and the Green function, allowing for the lack of smoothness of the boundary or of the coefficients of the equation. Building on our earlier account of history of the subject in [25], this survey presents the current state of the art, emphasizing the most recent results and emerging open problems. Contents 1991 Mathematics Subject Classification. Primary 35-02; Secondary 35B60, 35B65, 35J40, 35J55.
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