Local regularity and compactness for the p-harmonic map heat flows
-
Masashi Misawa
Abstract
We study a geometric analysis and local regularity for
the evolution of
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: 15K04962
Funding statement: The work of Masashi Misawa was partially supported by the Grant-in-Aid for Scientific Research (C) No. 15K04962 from Japan Society for the Promotion of Science.
A Appendix
Here we demonstrate the proof of Lemma 8,
based on Moser’s iteration method as usual.
Such an estimate has originally been done for the evolutionary
Proof of Lemma 8.
We use the same notation as in Lemma 8. The Bochner-type formula holds in
in the distribution sense
with
Here the positive constants
we now rewrite (A.1)
for the scaled solution
satisfying in
Formally make differentiation of equation (A.3) on spatial variable and multiply the resulting equation by the spatial gradient yielding
Here by (3.1) and Cauchy’s inequality we estimate the right-hand side as
leading to the scaled Bochner formula
for the scaled solution
Finally, we make Moser’s iteration estimate
by (A.4)
and scaling back to have (3.2).
Now, taking care of localization by the cut off function
Let
and also write as
Let
Applying the Sobolev embedding
which is combined with (A.5) and yields
By Hölder’s inequality and (A.6) we compute as
where we use a simple inequality valid for
and also estimate the derivative of
because by the range
and so we have the estimations
and
We arrange some terms in an appropriate way to have
Here let
and let
We choose
which is computed by sequences (A.8) and (A.9) as
An iterative application of (A.10) yields
with the relation of exponents
and the computation
Finally, scaling back in (A.11) yields the desired estimate (3.2). ∎
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© 2018 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Local regularity and compactness for the p-harmonic map heat flows
- 𝒟1,2(ℝN) versus C(ℝN) local minimizer on manifolds and multiple solutions for zero-mass equations in ℝN
- Existence and regularity results for weak solutions to (p,q)-elliptic systems in divergence form
- On a new class of fractional partial differential equations II
- On the structure of flat chains modulo p
- A remark on the two-phase obstacle-type problem for the p-Laplacian
Artikel in diesem Heft
- Frontmatter
- Local regularity and compactness for the p-harmonic map heat flows
- 𝒟1,2(ℝN) versus C(ℝN) local minimizer on manifolds and multiple solutions for zero-mass equations in ℝN
- Existence and regularity results for weak solutions to (p,q)-elliptic systems in divergence form
- On a new class of fractional partial differential equations II
- On the structure of flat chains modulo p
- A remark on the two-phase obstacle-type problem for the p-Laplacian
